Reading an article (A. Veneziani, T. Pereira, A note on the volume form in normal matrix space) I didn't understand this sentence:
"Given a unitary matrix $U$ we may use the representation $U=e^W$ where $W$ is a skew-hermitian matrix $W=-W^*$. For our case, W will be a skew-hermitian matrix which zero diagonal. This is to equalise the degrees of freedom of $U$."
I don't understand why we have to force the elements of the diagonals to become zero. The degrees of freedom of a $n*n$ unitary matrix are $n^2$ and the same is true for skew-hermitian matrix ($n(n-1)+n$, being the diagonal elements imaginary).
Oh, I understood! The author did not consider the unitary group, but the unitary group minus the n-torus. My bad sorry, thanks any way!