We know that there exists skew-symmetric unitary on $\mathcal{B(H)}$ when $\mathcal{H}$ is of even dimensions. In particular for $\mathcal{H}=\mathbb{C}^2$, any such matrix is scalar multiple of Pauli matrix $\sigma_y$. For higher dimensions, this is not the case though.
My question is, if we consider $\mathcal{H}$ to be an infinite dimension Hilbert space, does there exists such skew symmetric unitary operators. I have a feeling that they do exists, but could not manage to write the proof (or examples say for $L^2$ spaces). My motivation comes from partially quantum mechanics though. I thought $L^2$ spaces will be simpler spaces to handle before asking the same on a quantum system, or an abstract $C^*$ algebra.
I guess I am asking a stupid question. I did not get any good help after googling (perhaps I did not understand some of them) and so decided to ask it here. Advanced thanks for any help suggestion, references etc. Feel free to edit it or retag, if you think it is insufficient or wrongly tagged.