  Let $M$ be a random matrix in the orthogonal group $\O_n$, distributed
according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\R$
such that $\tr(AA^t)=n$. Then the total variation distance of the random
variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1)$, and this
rate is sharp up to the constant. Analogous results are obtained for $M$ a
random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$. The proofs
are applications of a new abstract normal approximation theorem which extends
Stein's method of exchangeable pairs to situations in which continuous
symmetries are present.
