Can we write any unitary matrix as the exponential of a skew-symmetric complex matrix

Question

Following the article https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Infinitesimal_rotations which states that a skew symmetric (or anti-symmetric) real matrix can be written as the exponential of a orthogonal matrix, I was wondering if every skew-symmetric complex matrix (that is, one that has complex components)can be written as the exponential of a unitary matrix.

EDIT:As pointed out in the answer below, I got it the other way around: I wanted to say that any orthogonal matrix can be written as the exponential of a skew-symmetric real matrix (which turns out that it does not hold for any orthogonal matrix)

Answer 1

It's the other way around: the exponential of every skew-symmetric matrix is an orthogonal matrix. And, no, not every orthogonal matrix can be obtained by this process; only those whose determinant is $1$.