Yes, you can show this using only the fact that the Clifford Algebra has a unique representation up to similarity transformation in any dimension. This is shown in the first few pages of

http://arxiv.org/pdf/hep-th/9811101.pdf

Then you observe that if $\gamma^\mu$ obeys the clifford algebra, then so does $-(\gamma^\mu)^T$. $\mathcal{C}$ is then defined as the similiarity transformation between the two representations, whose existence is guaranteed by the uniqueness of the representation of the Clifford algebra.

This post imported from StackExchange Physics at 2014-03-06 21:53 (UCT), posted by SE-user Dan