  We consider Kac's random walk on $n$-dimensional rotation matrices, where
each step is a random rotation in the plane generated by two randomly picked
coordinates. We show that this process converges to the Haar measure on
$\mathit{SO}(n)$ in the $L^2$ transportation cost (Wasserstein) metric in
$O(n^2\ln n)$ steps. We also prove that our bound is at most a $O(\ln n)$
factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and
Pak/Sidenko, had extra powers of $n$ and held only for $L^1$ transportation
cost. Our proof method includes a general result of independent interest, akin
to the path coupling method of Bubley and Dyer. Suppose that $P$ is a Markov
chain on a Polish length space $(M,d)$ and that for all $x,y\in M$ with
$d(x,y)\ll1$ there is a coupling $(X,Y)$ of one step of $P$ from $x$ and $y$
(resp.) that contracts distances by a $(\xi+o(1))$ factor on average. Then the
map $\mu\mapsto\mu P$ is $\xi$-contracting in the transportation cost metric.
