  We consider an analogue of the Kac random walk on the special orthogonal
group $SO(N)$, in which at each step a random rotation is performed in a
randomly chosen 2-plane of $\bR^N$. We obtain sharp asymptotics for the rate of
convergence in total variance distance, establishing a cut-off phenomenon in
the large $N$ limit. In the special case where the angle of rotation is
deterministic this confirms a conjecture of Rosenthal \cite{Rosenthal}. Under
mild conditions we also establish a cut-off for convergence of the walk to
stationarity under the $L^2$ norm. Depending on the distribution of the
randomly chosen angle of rotation, several surprising features emerge. For
instance, it is sometimes the case that the mixing times differ in the total
variation and $L^2$ norms. Our estimates use an integral representation of the
characters of the special orthogonal group together with saddle point analysis.
