  Several moduli spaces parametrizing linear subspaces of the projective space
are cut out by linear and quadratic equations in their natural embedding:
Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper
is to prove that a similar statement holds when one replaces linear subspaces
with algebraic subschemes of the projective space. We exhibit equations of
degree 1 and 2 that define schematically the Hilbert schemes
$\mathbf{Hilb}^{p}_{\mathbb P^n}$ for all (possibly nonconstant) Hilbert
polynomials $p$. The equations are reminiscent of the Pl\"ucker relations on
the Grassmannians: they are built formally with permutations on indexes on the
Pl\"ucker coordinates. Our method relies on a new construction of the Hilbert
scheme as a quotient of a scheme of quiver representations.
