Chow classes of orbit closures of skew-symmetric matrix pencils

Let $V$ be a six-dimensional complex vector space and let $G=\mathrm{Gr}(2,{\mathrm{\Lambda }}^2{V}^{\vee })$ parametrize pencils of skew-symmetric $6\times 6$ matrices. The group $\mathrm{PGL}(V)$ has $11$ orbits on $G$, classified by the Jordan-Kronecker canonical form. We compute the Chow class of every orbit closure in the Schubert basis of ${\mathrm{CH}}^{\ast }(G)$.