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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?

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

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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 $\Bbb C$.

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