In this paper, Debarre-Voisin refer to the "moduli space" of differential 3-forms $\sigma \in \bigwedge^3(V_{10}^*)$ on a fixed vector space $V_{10}$ of dimension 10, and state that this space is $\mathbb{P}\left( \bigwedge^3(V_{10}^*) \right) // PGL(V_{10}^*)$. The authors also suggest that "a dimension count" reveals that the dimension of this moduli space is 20. (I suppose that the double slash refers to a GIT quotient.)
Is there a relatively simple way of understanding what this space is, and why exactly it parameterizes differential 3-forms on $V_{10}$? What about its dimension?