This might be a stupid question due to my lack of knowledge in algebraic geometry. So I have a subvariety $V\subset G_k(\mathbb{C}^n)$ of codimension $k$. The only thing I know about $V$ is its cohomology class Poincare - dual to its fundamental class $[V] \in H_*(G_k(\mathbb{C}^n))$ and its codimension. That is, I have $$[V]^* := PD^{-1}([V]) \in H^{2r}(G_k(\mathbb{C}^n)),$$ where $r =\text{codim}(V)$.
If $V$ was a subvariety of the projective space $\mathbb{P}^N$, I would take the cup - product of $[V]^*$ with the (Schubert - ) class $c_1(\mathcal{O}_{\mathbb{P}^N}(1))^{N-r} = \sigma_1^{N-r}$ of a generic $r$ - plane in $\mathbb{P}^N$ to get its degree (right?).
But since the Grassmannian $G_k(\mathbb{C}^n)$ is only embedded into some $\mathbb{P}^N$ via Plücker - embedding, I don't know what the "class of a generic $r$ - plane" in the Grassmannian looks like.
Thank you in advance!