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Is there a skew-symmetric matrix that has at least one eigenvalue with multiplicity greater than one?

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Take a nonzero skew symmetric matrix $A$; this has a nonzero eigenvalue (over $\Bbb C$). Then build a block matrix $B=\pmatrix{A&0\\0&A}$. This is skew-symmetric and its eigenvalues are the same as those of $A$ but with twice the multiplicity.

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Take the matrix$$\begin{bmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}.$$The eigenvalues $\pm i$ have both multiplicity $2$.

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