Freedoms of real orthogonal matrices

I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric matrix yields an orthogonal matrix:

$M^T=-M, c=\exp(M) \Rightarrow c^T=c^{-1}$

A antisymmetric matrix possesses $\frac{n(n-1)}{2}$ degrees of freedom.

BUT: When I also thought about how to parametrize these freedoms explicitly (without the exponential) I remembered, that rotations in $\mathbb{R}^n$ can be parametrized using $n-1$ angles or cosines.

I dont’ understand, where the remaining parameters are hidden?

My guess is, that a orthoganl transformation in $n>3$ can be more complicated than a rotation or that there are different types of rotation (containing reflectiong or such things) and that all the combination of these different types accounts for the rest of the parameters.

Thanks you for your help!

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The result you remembered is wrong. I am actually not sure what you are referring to. Maybe you are thinking about Euler angles in $3$ dimensions, which are terribly misleading. Each Euler angle should not be thought of as a rotation about a different axis, but as a rotation in a different $2$-dimensional subspace spanned by two axes. The fact that these are the same thing in $3$ dimensions is a consequence of the anomalous identity ${3 \choose 1} = {3 \choose 2}$ (equivalently, the existence of the cross product) and does not hold in higher dimensions.

For example, in four dimensions there are four coordinate axes, say $x, y, z, w$, and ${4 \choose 2} = 6$ Euler angles: $xy$-rotation, $xz$-rotation, $xw$-rotation, $yz$-rotation, $yw$-rotation, $zw$-rotation. So the correct version of this result really does give the correct number of parameters.

Although every rotation will have an matrix which is orthogonal, not every orthogonal matrix is the matrix of a rotation. And you need more than just the angles in the general case to specify a rotation.

In fact, every orthogonal matrix is similar to a block-diagonal matrix, each block $2\times 2$ and at most a $1\times 1$ block; the $2\times 2$ blocks correspond to rotations, and the $1\times 1$ block corresponds either to $1$ or to $-1$ (an identity or a reflection about a codimension $1$ subspace).

Each rotation can be determined by a single angle, (and the $1\times 1$ block gives you a furhter degree of freedom, but this does not actually give you a full fledged “dimension”). But you need more than the angles to specify the rotations, you need to specify the $2$-dimensional subspaces in which they act.

In $\mathbb{R}^3$, you can get away with just three angles because you can determine a single vector with two angles (think spherical coordinates); this gives you the normal vector to the plane of rotation; then specify the angle of rotation, and this gives you the $3 = \frac{3(2)}{2}$ parameters.

But in $\mathbb{R}^n$ with $n\gt 3$, you need to specify the $2$-dimensional subspace and the angle to specify the rotation (and do this enough times). This cannot be determined by a single vector plus an angle any more the way it does in $\mathbb{R}^3$, because these planes are no longer codimension $1$ (they cannot be given as “the orthogonal complement to this one-dimensional subspace).

To describe a rotation in $\mathbb{R}^n$ on needs $n(n-1)/2$ parameters, which equals the number of freedoms that a orthogonal matrix possesses.

(To describe a direction in $\mathbb{R}^n$ one needs $n-1$ angles. See