It is a known fact that if $X$ is a skew-symmetric matrix, then $e^X$ is an orthogonal matrix.
Is it also true the opposite, ie that any orthogonal matrix admits a representation like $e^X$ for some $X$ skew-symmetric?
If not, is there a way to parametrize the space of the orthogonal matrices? With this I mean, a function $f$ that takes some parameters $\Theta$ and returns a matrix $O$ such that:
- $f(\Theta)=O$ is orthogonal
- If $O$ is orthogonal, there exists a set of parameters $\Theta$ s.t. $f(\Theta)=O$?
Does the topic get easier if we focus on orthonormal matrices?