The matrix of $U$ in $\mathbb{R^3}$,with the standard inner product which is rotation of the plane $W=sp\{\alpha_1,\alpha_2\}$ about the orthogonal line $\alpha_3$ through the angle $\theta$ , where $\alpha_1,\alpha_2$ are orthonormal basis of $W$ and $\alpha_3$ is the vector of norm $1$ which is orthogonal to $W$.
Since $\alpha_3$ is a vector of norm $1$ and the matrix of rotation is about the line $sp\{\alpha_3\}$.
Then, $U(\alpha_3)=\alpha_3$.
Also,$U(\alpha_1)=cos(\theta).\alpha_1-sin(\theta).(\alpha_2)+0.\alpha_3$.
$U(\alpha_2)=sin(\theta).\alpha_1+cos(\theta).\alpha_2+0.\alpha_3$
Although, I cant reason out the last two equations but this is whatmy intuition.I need help after this.