$U = \mathrm{e}^{i \theta}$. What does it do geometrically to a plane $\mathbb{R}^2$?

Question

Let $i$ to be a matrix such that $$i = \begin{bmatrix} 0 & 1 \\ -1& 0 \end{bmatrix}.$$ Then what does $ U = \mathrm{e}^{i \theta}$ do to a plane $\mathbb{R}^2$?

I think it's a rotation, but I struggle to justify it or disprove it.