A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like a norm. Also, the square root of this number has a geometrical interpretation as the scale factor by which $L$ maps $n$-volumes, which makes Grassmann’s exterior product come into mind.

Question If we let $\Lambda^n(\mathbb{R}^N)$ denote the vector space spanned by $n$-vectors

$$v_1\wedge \ldots \wedge v_n,\qquad v_j \in \mathbb{R}^N,$$

does there exist a scalar product $\langle ,\rangle$ on it such that
$$\det(L^TL)=\langle Le_1\wedge \ldots \wedge Le_n, Le_1\wedge \ldots \wedge Le_n\rangle ?^{(\star)}$$

If the answer is affirmative, is this scalar product geometrically related to the concept of
“oriented $n$-volume in $\mathbb{R}^N$”? And finally, is it possible to
generalize all this to an arbitrary Riemannian manifold?

Bibliographical references as answers are fine. Thank you.

(*) $e_1\ldots e_n$ denotes the standard basis of $\mathbb{R}^n$.

Solutions Collecting From Web of "A scalar product in the space of oriented volumes?"

See section 9.4 Differential Forms and Metrics in Jeff Lee’s Manifolds and Differential Geometry. He explains precisely how an inner product on a space V induces an inner product on the exterior algebra of V. I think you will find his presentation enlightening.