In Example 1.2.22 of [these notes][1]  you will find a description of the manifold structure  of $\DeclareMathOperator{\Gr}{Gr}$ $\newcommand{\bC}{\mathbb{C}}$ $\Gr_n(V)$,  where $V$ is a finite dimensional complex Hilbert space. The finite dimensionality is used only in computing the dimension.  It is described  as a submanifold in the real  Hilbert space of (bounded) self-adjoint operators  $V\to V$.  As such it has  an induced metric.

If  you  fix a  orthonormal basis $(e_n)_{n\geq 1}$ of $V$, then you can produce  a stratification of $\Gr_k(V)$ by Schubert cells   of finite codimension. One can show that these define cohomology classes spanning the cohomology  of $\Gr_k(V)$;   Appendix A of [this paper of Daniel  Cibotaru][2]    explains how one can  associate cohomology classes to strata under certain conditions that are satisfied in this case.

These strata also have a Morse theoretic description.




  [1]: https://www3.nd.edu/~lnicolae/Lectures.pdf
  [2]: https://arxiv.org/pdf/0901.2563.pdf