# Skew-symmetric non-diagonalizable matrix

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?

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 $$\Bbb C$$.