Here is a slightly different way to do this: the cohomology of $G(k,n)$ is generated by the Chern classes $c_1,\ldots,c_k$ of the universal bundle $\gamma (k,n)$ and the Chern classes $c'_1,\ldots,c'_{n-k}$ of the universal quotient bundle, subject to one relation,

$$(1+c_1+\cdots +c_k)(1+c_1'+\cdots +c'_{n-k})=1.$$

Using this we can express $c'_i$'s in terrms of $c_i$'s or vice versa.

Now if we embed $G(k_1,n_1)\times G(k_2,n_2)$ in $G(k,n)$ as above $\gamma(k,n)$ restricts to $p^*_1(\gamma(k_1,n_1))\oplus p^*_2(\gamma(k_2,n_2))$ where $p_1$ and $p_2$ are the projections to the first, respectively, second factor. This allows one to compute the cohomology map $H^*(G(k,n))\to H^*(G(k_1,n_1)\times G(k_2,n_2))$ induced by the embedding, and hence also the homology map in the opposite direction. This gives an explicit way of calculating the image of the fundamental class modulo the values of the Chern classes on the Schubert cells.

(If necessary I can add more details.)