Paired eigenspaces of the square of a real skew-symmetric matrix ($\max \#\mathrm{spec}(M^2)=\lceil n/2\rceil$)

For a real skew-symmetric matrix $M$, $M^2$ is symmetric and $x^T{M}^{2}x=-\Vert Mx{\Vert }^2$. On an eigenspace with eigenvalue $-{\mu }^2<0$, every unit vector $u$ has unit partner $v=Mu/\mu$, orthogonal to $u$, with $Mu=\mu v$ and $Mv=-\mu u$. Its plane and its orthogonal complement are invariant under $M$. Repeating gives real orthogonal two-dimensional blocks $\left(\begin{array}{cc}0& -\mu \\ \mu & 0\end{array}\right)$ and a zero block. Thus every nonzero eigenvalue of $M^2$ has even multiplicity, the zero multiplicity has the parity of $n$, and the number of distinct eigenvalues is at most $n/2$ for even $n$ and $(n+1)/2$ for odd $n$. Distinct positive block parameters attain the bound, with one zero direction added when $n$ is odd.