  Let $K$ be a field of characteristic different from 2 and let $C$ be an
octonion algebra over $K$. We show that there is a seven-dimensional subspace
of $7\times 7$ skew-symmetric matrices over $K$ which is invariant under the
automorphism group of $C$. This subspace consists of elements of rank 6 when
$C$ is a division algebra, and elements of rank 4 and 6 when $C$ is a split
algebra. In the latter case, the automorphism group is the exceptional group
$G_2(K)$.
