Given any skew-symmetric n x n matrix A, we have

det (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),

whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A 2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A 2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A 2 - λI) of A 2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).