Let $U \in \mathbb{R}^{n \times n}$ be a unitary matrix, $U$ can be nonsymmetric, its eigenvalues can be complex numbers and all have modulus $1$.

Is there an upper bound for the maximum **singular value** of its skew symmetric part (which is not necessarily unitary) depending on its eigenvalues?

i.e.: Is there an $f$ such that $\left\|\frac{U - U^T}{2} \right\|_2 = \sigma_\text{max}\left(\frac{U - U^T}{2}\right) \le f\left(\lambda_i\left(U\right)\right)$ ?

More details:

Observe that if $U=I$ (eigenvalues are real) $\Rightarrow \left\|\frac{U - U^T}{2} \right\|_2 = \sigma_\text{max}\left(\frac{U - U^T}{2}\right) = 0$, and if $U$ is skew-symmetric (eigenvalues purely imaginary) $\Rightarrow\left\|\frac{U - U^T}{2} \right\|_2 = \sigma_\text{max}\left(\frac{U - U^T}{2}\right) = 1$. Therefore there is a relationship between the norm $\left\|\frac{U - U^T}{2} \right\|_2 = \sigma_\text{max}\left(\frac{U - U^T}{2}\right)$ and the argument of the eigenvalues of $U$, i.e. $f\left(\lambda_i\left(U\right)\right) = f\left(\text{arg}(\lambda_i\left(U\right))\right)$.

Further notes: in my work $U$ is the unitary factor of the polar decomposition of an M-matrix, but this may be irrelevant.