Pseudo-orthogonal eigenvalues of skew-symmetric matrices

The following result is attributed to J. Williamson: Every real, symmetric, and positive definite matrix $A$ of even order $n = 2m$ can be brought to diagonal form by congruence with a symplectic transformation matrix. The diagonal entries of this form are invariants of congruence transformations performed with $A$ and are called the symplectic eigenvalues of this matrix. In this short paper, we prove an analogous fact concerning (complex) skew-symmetric matrices and transformations belonging to a different group, namely, the group of pseudo-orthogonal matrices.