  We give a basis for the vector space generated by rational Hodge classes of
type (2,2) on the Hilbert square of a projective K3 surface general in its
rank, which is a subspace of the singular cohomology ring with rational
coefficients: we use Nakajima operators and an algebraic model developed by
Lehn and Sorger as main tools. We then obtain a basis of the lattice generated
by integral Hodge classes of type (2,2) on the Hilbert square of a projective
K3 surface general in its rank: we exploit lattice theory, a theorem by Qin and
Wang and a result by Ellingsrud, G\"ottsche and Lehn.
