In a recent paper ([1]), Ravichandran and Srivastava study pavings for collections of matrices. Their main theorem yields an improvement to the bound obtained by Johnson, Ozawa, and Schechtman (noted in Bill Johnson's answer above). In particular, Ravichandran and Srivastava's work implies the following:
Corollary (Corollary 3 in [1]). Every zero trace matrix $A \in M_n(\mathbb{C})$ may be written as $A=[B,C]$ such that $\|B\|$, $\|C\| \le K\log^2(n)\|A\|$ for some universal constant $K$.
[1]. M. Ravichandran and N. Srivastava. Asymptotically Optimal Multi-Paving. arXiv. Jun 2017.