Some problems with Exponential matrix

Question

We've know that $e^A$ is Orthogonal when $A$ is Real anti-symmetric, how can we know that for any orthogonal matrix we can find the real anti-symmetric matrix. Similarly, for one unitary matrix $U$, is there a skew-Hermitian matrix $A$ which makes $e^A = U$?

Answer 1

We can't. The set of matrices $e^A$ where $A$ is real antisymmetric is connected, but the set of orthogonal matrices isn't: some have determinant $1$ and others have determinant $-1$.

In the unitary case, take $A = \log(U)$ (using the holomorphic functional calculus) where $\log$ is a branch of the logarithm that is analytic in a neighbourhood of the spectrum of $A$.