Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$.
The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$,
and clearly the 7-dimensional representation of $G_2$ in $V$ is isomorphic to its representation in the space of pure octonions.

 I know from a classification of alternating trilinear forms in dimension 7 that there exists a $G_2$-invariant alternating trilinear form
 $\omega\in\Lambda^3 V^*$

>  **Question.** What is a coordinate-free description of a $G_2$-invariant alternating trilinear form on $V$?