Over $\mathbb R$ or $\mathbb C$, a square matrix is skew-symmetric if $A^T=-A$. Its diagonal is zero, and $x^TAx=0$ for every vector $x$. Every square matrix decomposes uniquely as $(A+A^T)/2+(A-A^T)/2$, a symmetric plus skew-symmetric part.
For a real skew-symmetric matrix, complex eigenvalues are purely imaginary or zero. Over a field of characteristic two, the zero-diagonal conclusion and division-by-two decomposition need modification.
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