We exhibit cocycles representing certain classes in the cohomology of the algebraic group ${\mathrm{GL}}_n$ with coefficients in the representation ${\Gamma }^{*}({\mathfrak{gl}}_n^{(1)})$. These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras (see [18])