On the Lefschetz locus in Gor(1,n,n,1)

We study two special families of cubic hypersurfaces with vanishing Hessian in ${\mathbb{P}}^N$, obtaining rational parametrizations and computing their degree in $\mathbb{P}(S_3)$. For $N\le 6$, these two families exhaust the locus of cubics with vanishing Hessian that are not cones. As a consequence, via Macaulay-Matlis duality, we obtain a description of the locus in $\mathrm{G}\mathrm{o}\mathrm{r}(1,n,n,1)$ corresponding to those algebras that satisfy the Strong Lefschetz property, for $n\le 7$.