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!