Are Skew Symmetric Matrices Diagonalizable?

Since a real skew-symmetric matrix is normal, it is diagonalizable (by a unitary matrix).

Is skew Hermitian matrix diagonalizable?

Properties. The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.

Is skew-symmetric matrix Nonsingular?

The result implies that every odd degree skew-symmetric matrix is not invertible, or equivalently singular. Also, this means that each odd degree skew-symmetric matrix has the eigenvalue 0.