Cross-product matrix
= Cross-product matrix
For $a=(a_1,a_2,a_3)^T$, the map $b\mapsto a\times b$ has matrix
$$
[a]_\times=
\begin{pmatrix}
0&-a_3&a_2\\
a_3&0&-a_1\\
-a_2&a_1&0
\end{pmatrix}.
$$
It has trace zero and annihilates $a$, so its determinant is zero.