I could find many resources on generating random unitary matrices, usually citing F. Mezzadri, *Notices of the AMS* 54 (2007), 592-604 for a method which generates unitaries random with respect to the Haar measure, but none of them mentioned the special unitary subgroup. In particular, I'm not sure about one thing: say that we take random $n\times n$ unitaries $U_n$ generated this way and construct special unitary matrices simply by $U_n / \det(U_n)^{1/n}$.

Are special unitary matrices generated this way also random with respect to the Haar measure?