$o(n) =$ Set of all real skew-symmetric matrices $O(n) =$ Set of all real orthogonal matrices
We know that whenever $X\in o(n)$ , then $e^X\in O(n)$. Also, I know that the map $g:o(n)\to O(n)$ such that $g(X)=e^X$ is a continuous map.
However, I'm unable to figure out whether this map is surjective or not, i.e. for every $Y\in O(n)$ , does $\exists$ some $X\in o(n)$ such that $Y=e^X$ ? Also, is this map necessarily injective ?
Thanks in advance. Sorry if this question is silly or previously answered.
