Is Skew Symmetric Matrix Diagonalizable?

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

Are all symmetric matrices diagonalizable?

Real symmetric matrices not only have real eigenvalues, they are always diagonalizable. In fact, more can be said about the diagonalization.

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.