= Solution
The original PDF requires $B$ itself to be an <orthogonal skew-symmetric matrix>; it does not require $C$ to be diagonal. A real two-dimensional <skew-symmetric matrix> has the form $B=\begin{pmatrix}0&u\\-u&0\end{pmatrix}$. The <orthogonal matrix> condition gives $u^2=1$, so choose $u=1$. Then $C=B^{\mathsf T}AB$ is forced:
$$
\boxed{B=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad C=\begin{pmatrix}-4&-3\\-3&4\end{pmatrix}.}
$$
Direct multiplication verifies $B^{\mathsf T}=-B$, $B^{\mathsf T}B=I$ and $BCB^{\mathsf T}=A$. Choosing $-B$ gives the same $C$. Replacing $B$ by a matrix of <eigenvectors> to make $C$ diagonal would generally fail the printed antisymmetry requirement.
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