= Orthogonal skew-symmetric matrix
{title2=$B^{\mathsf T}B=I,\quad B^{\mathsf T}=-B\iff B^2=-I$}
= Skew-symmetric orthogonal matrix
{synonym}
A real <skew-symmetric matrix> is an <orthogonal matrix> exactly when its square is minus the <identity matrix>. Indeed, $B^{\mathsf T}B=-B^2$. Such a <matrix> can exist only in even <dimension>: taking <determinants> of $B^2=-I$ would otherwise give $(\det B)^2=-1$. In two dimensions every example is
$$
B=\pm\begin{pmatrix}0&1\\-1&0\end{pmatrix}.
$$
These are <rotation matrices> through opposite right angles. Their <orthogonal similarity> transforms preserve <eigenvalues>, but do not in general produce a <diagonal matrix>.
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