Let $\mathcal{H}$ be an infinite dimensional separable (complex) Hilbert space. What is a natural space which parameterizes the choices of orthonormal bases for $\mathcal{H}$?

It seems like one option, in analogy to the finite-dimensional Stiefel manifold (something I know nothing about beyond what I just glanced at on the wikipedia page), would be to consider the set of all unitary operators from $\ell^2(\mathbb{N})\to\mathcal{H}$, with some topology. What kind of topology would one consider? Ideally, I'd like it to be Polish.