Skew-symmetric non-diagonalizable matrix

Question

Do you have an example of a real skew-symmetric matrix (seen as an operator over $\mathbb{C}^n$) having at least one (purely imaginary) eigenvalue with algebraic multiplicity strictly greater than the geometric one?

Answer 1

Such a matrix doesn't exist. Since it is skew-symmetric, it's normal and therefore diagonalizable.

Answer 2

Every real skew-symmetric matrix is conjugate over $\Bbb R$ to a diagonal sum of matrices of the forms $\pmatrix{0}$ and $\pmatrix{0&a\\-a&0}$ and so is diagonalisable over $\Bbb C$.