In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent or coordinate independent, or at least, changes in some covariant way under changes of basis or coordinates. One way to ensure that our structures adhere to this principle is to give their definitions without reference to a basis. Often we employ universal properties, functors, and natural transformations to encode these natural, coordinate/basis free structures. But the Riemannian volume form does not appear to admit such a description, nor does its pointwise analogue in linear algebra.
Let me list several examples.
In linear algebra, an inner product on $V$ is an element of $\operatorname{Sym}^2{V^*}$. The symmetric power is a space which may be defined by a universal property, and constructed via a quotient of a tensor product. No choice of basis necessary. Alternatively an inner product can be given by an $n\times n$ symmetric matrix. The correspondence between the two alternatives is given by $g_{ij}=g(e_i,e_j)$. Calculations are easy with this formulation, but one should check (or require) that the matrix transforms appropriately under changes of basis.
In linear algebra, a volume form is an element of $\Lambda^n(V^*)$. Alternatively one may define a volume form operator as the determinant of the matrix of the components of $n$ vectors, relative to some basis.
In linear algebra, an orientation is an element of $\Lambda^n(V^*)/\mathbb{R}^>$.
In linear algebra, a symplectic form is an element of $\Lambda^2(V^*)$. Alternatively may be given as some $\omega_{ij}\,dx^i\wedge dx^j$.
In linear algebra, given a symplectic form, a canonical volume form may be chosen as $\operatorname{vol}=\omega^n$. This operation can be described as a natural transformation $\Lambda^2\to\Lambda^n$. That is, to each vector space $V$, we have a map $\Lambda^2(V)\to\Lambda^n(V)$ taking $\omega\mapsto \omega^n$ and this map commutes with linear maps between spaces.
In differential geometry, all the above linear algebra concepts may be specified pointwise. Any smooth functor of vector spaces may be applied to the tangent bundle to give a smooth vector bundle. Thus a Riemannian metric is a section of the bundle $\operatorname{Sym}^2{T^*M}$, etc. A symplectic form is a section of the bundle $\Lambda^2(M)$, and the wedge product extends to an operation on sections, and gives a symplectic manifold a volume form. This is a global operation; this definition of a Riemannian metric gives a smoothly varying inner product on every tangent space of the manifold, even if the manifold is not covered by a single coordinate patch
In differential geometry, sometimes vectors are defined as $n$-tuples which transform as $v^i\to \tilde{v}^j\frac{\partial x^i}{\partial \tilde{x}^j}$ under a change of coordinates $x \to \tilde{x}$. But a more invariant definition is to say a vector is a derivation of the algebra of smooth functions. Cotangent vectors can be defined with a slightly different transformation rule, or else invariantly as the dual space to the tangent vectors. Similar remarks hold for higher rank tensors.
In differential geometry, one defines a connection on a bundle. The local coordinates definition makes it appear to be a tensor, but it does not behave the transformation rules set forth above. It’s only clear why when one sees the invariant definition.
In differential geometry, there is a derivation on the exterior algebra called the exterior derivative. It may be defined as $d\sigma = \partial_j\sigma_I\,dx^j\wedge dx^I$ in local coordinates, or better via an invariant formula $d\sigma(v_1,\dotsc,v_n) = \sum_i(-1)^iv_i(\sigma(v_1,\dotsc,\hat{v_i},\dotsc,v_n)) + \sum_{i+j}(-1)^{i+j}\sigma([v_i,v_j],v_1,\dotsc,\hat{v_i},\dotsc,\hat{v_j},\dotsc,v_n)$
Finally, the volume form on an oriented inner product space (or volume density on an inner product space) in linear algebra, and its counterpart the Riemannian volume form on an oriented Riemannian manifold (or volume density form on a Riemannian manifold) in differential geometry. Unlike the above examples which all admit global basis-free/coordinate-free definitions, we can define it only in a single coordinate patch or basis at a time, and glue together to obtain a globally defined structure. There are two definitions seen in the literature:
Does anyone else think that one of these definitions sticks out like a sore thumb? Does it bother anyone else that in linear algebra, the volume form on an oriented inner product space doesn’t exist as natural transformation $\operatorname{Sym}^2 \to \Lambda^n$? Do the instructions to “take the determinant of a bilinear form” scream out to anyone else that we’re doing it wrong? Does it bother anyone else that in Riemannian geometry, in stark contrast to the superficially similar symplectic case, the volume form cannot be defined using invariant terminology for the whole manifold, but rather requires one to break the manifold into patches, and choose a basis for each? Is there any other structure in linear algebra or differential geometry which suffers from this defect?
Answer: I’ve accepted Willie Wong’s answer below, but let me also sum it up, since it’s spread across several different places. There is a canonical construction of the Riemannian volume form on an oriented vector space, or pseudoform on a vector space. At the level of level of vector spaces, we may define an inner product on the dual space $V^*$ by $\tilde{g}(\sigma,\tau)=g(u,v)$ where $u,v$ are the dual vectors to $\sigma,\tau$ under the isomorphism between $V,V^*$ induced by $g$ (which is nondegenerate). Then extend $\tilde{g}$ to $\bigotimes^k V^*$ by defining $\hat{g}(a\otimes b\otimes c,\dotsb,x\otimes y\otimes z\dotsb)=\tilde{g}(a,x)\tilde{g}(b,y)\tilde{g}(c,z)\dotsb$. Then the space of alternating forms may be viewed as a subspace of $\bigotimes^k V^*$, and so inherits an inner product as well (note, however that while the alternating map may be defined canonically, there are varying normalization conventions which do not affect the kernel. I.e. $v\wedge w = k! Alt(v\otimes w)$ or $v\wedge w = Alt(v\otimes w)$). Then $\hat{g}(a\wedge b\dotsb,x\wedge y\dotsb)=\det[\tilde{g}(a,x)\dotsc]$ (with perhaps a normalization factor required here, depending on how Alt was defined).
Thus $g$ extends to an inner product on $\Lambda^n(V^*)$, which is a 1 dimensional space, so there are only two unit vectors, and if $V$ is oriented, there is a canonical choice of volume form. And in any event, there is a canonical pseudoform.
A few points:
Let me just summarise by giving the punch line again:
For every inner product $g$ on a vector space $V$ there are two compatible volume forms in $\Lambda^n V$: they differ by sign. Therefore the natural mapping from inner products takes image in $\Lambda^n V / \pm$!
Therefore if you want to construct a map based on fibre-wise operations on $TM$ sending Riemannian metrics to volume forms, you run the very real risk that, due to the above ambiguity, what you construct is not even continuous anywhere. The “coordinate patch” definition has the advantage that it sweeps this problem under the rug by implicitly choosing one of the two admissible local (in the sense of open charts) orientation. You can do without the coordinate patch if you start, instead, with an orientable Riemannian manifold $(M,g,\omega)$ and use $\omega$ to continuously choose one of the two admissible pointwise forms.
A coordinate-free definition of volume form is in fact well-known and frequently used, e.g. the cited Wikipedia article. I will try to reproduce it the nutshell to the best of my understanding.
Let $V$ be a (real, for certainty) vector space of finite dimension $\dim V = n$. The space of $n$-forms $\Lambda^n (V)$ has dimension 1. Thus $\Lambda^n (V)$ isomorphic to $\mathbb{R}$, however this isomorphism is not canonical: any choice of non-trivial $n$-form $\omega$ can be mapped to $1 \in \mathbb{R}$.
A volume form on a finite-dimensional vector space $V$ is a choice of a top-rank non-trivial exterior form (skew-symmetric $n$-linear functional) $\omega \in \Lambda^n (V)$. I think that this definition is quite coordinate-free.
Once such a form has been chosen, it can be used to divide the space of bases in $V$ into two classes that are called orientations. There are two of them, positive ($\omega > 0$) and negative ($\omega < 0$). Having a volume form chosen, one can speak about oriented volumes of parallelotopes, for instance.
If for any reason we have an inner product $g$ in $V$ we can make this choice canonical. One needs to consider orthonormal frames (with respect to $g$). The canonical volume form will take value 1 on positively oriented orthonormal frames.
The volume form of an inner-product space $(V, g)$ is that canonical choice of a volume form. It can be denoted by $Vol_{g}$ provided one also keeps in mind that there is a choice of orientation involved.
Along these lines one can obtain an understanding of the volume form as the Hodge dual of 1 in a pretty coordinate-free manner.
The metric tensor is symmetric and so can be orthogonally diagonalized, which means that its determinant is just the product, with multiplicity, of its eigenvalues. If we rotate the entire chart such that the coordinate axes at the point of interest follow the the eigenspaces, this means that $|g|$ tells us the factor that relates the squared coordinate length of each side of an infinitesimal box with its squared intrinsic length, all multiplied together. Therefore, multiplying with the square root of the determinant will convert a coordinate volumne of an infinitesimal box to the intrinsic volume.
The intuition to get here is that a Riemannian manifold, by definition, locally looks like Euclidean space, where the volume of a rectangular box is the product of its side lengths, and we can measure the volume of some particular figure with boxes of any particular orientation and get the same result. We can do this in a manifold too, since the orthogonality of the sides of a box, as well as its side lengths, are all intrinsic. The volume form simply expresses how to do this given any particular chart, by — for convenience — using boxes of an orientation that fits with the eigendecomposition of the metric. But the result must be invariant, because the volume of a piece of Euclidean space does not depend on how we turn the coordinate system.