  Motivated by the study of Killing forms on compact Riemannian manifolds of
negative sectional curvature, we introduce the notion of generalized vector
cross products on $\mathbb{R}^n$ and give their classification. Using previous
results about Killing tensors on negatively curved manifolds and a new
characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose
associated $3$-form is Killing, we then show that every Killing $3$-form on a
compact $n$-dimensional Riemannian manifold with negative sectional curvature
vanishes if $n\ge 4$.
