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Let $K$ be a field of characteristic different from 2 and let $C$ be anoctonion algebra over $K$. We show that there is a seven-dimensional subspaceof $7\times 7$ skew-symmetric matrices over $K$ which is invariant under theautomorphism 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 splitalgebra. In the latter case, the automorphism group is the exceptional group$G_2(K)$.

Subspaces of 7 x 7 skew-symmetric matrices related to the group G_2