If $X\subset \mathbb{P}^{n}$ is a cubic hypersurface that is not normal, what's the easiest way to see that the nonnormal locus is a linear subspace of dimension $n-2$?

As for a reference, there is a paper that classifies the nonnormal cubic hypersurfaces that gives a proof by expressing the hypersurface as a cone over the projection of a rational normal scroll of degree 3 (http://www.sciencedirect.com/science/article/pii/S0022404910002872), but just the fact about the nonnormal locus was already stated an earlier paper without proof (third paragraph of page 6 of https://arxiv.org/pdf/math/0005146v1.pdf), which makes me guess there might an easy way to prove this fact if we aren't after a classification.