Edit: I correct the mistake pointed out by Matthew in his comment.

In fact, any matrix $X$ can be written as $X=\lambda(U+V)$ for unitary matrices $U,V$, and $\lambda$ can be taken as a half of the operator norm of $X$. For a proof, see [this][1] question. This is slightly stronger than Matthew's comment, and the proof works in any **finite** von Neumann algebra (i.e. when the partial isometry in the polar decomposition can be taken as a unitary). In a $C^*$-algebra this is not possible (consider $z \mapsto z$ in the $C^*$-algebra of continuous functions on the unit disc of the complex plane) but, as noted by Matthew, you can get the same decomposition with $4$ unitaries.

  [1]: http://mathoverflow.net/questions/50936