Every real, skew-symmetric matrix is diagonalisable by a unitary matrix

Question

I need to show, that every real, skew-symmetric matrix M can be diagonalized by a unitary matrix U. $$ M=-M^T \implies M = U D U^\dagger \quad \textrm{with} \quad U U^\dagger = U^\dagger U = \mathbb{I} $$ I managed to show that $D$ is purely imaginary and that $U^\dagger U$ is a real, symmetric matrix, but i can't quite get to $U^\dagger U = \mathbb{I}$.