I assume that the norm is the spectral norm.

By the polar decomposition, any unit norm matrix can be written as $UD$ where $U$ is unitary and $D$ is matrix which, in some basis $E$, is diagonal with non-negative entries not greater than $1$. The diagonal entries can thus be written as $\cos(\theta_1),...,\cos(\theta_n)$ for real $\theta_i$'s. $UD$ is then the average of $U V$ and $U V^*$ where $V$ is the matrix which, in the basis $E$, is diagonal with diagonal entries $e^{i\theta_1},...,e^{i\theta_n}$.