  Let $S$ be the first degeneracy locus of a morphism of vector bundles
corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove
that, under certain positivity conditions, its Hilbert square
$\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an
irreducible homogeneous vector bundle on a product of Grassmannians. Our
construction involves a naturally associated Fano variety, and an explicit
description of the isomorphism.
