We study the properties of the multivariate skew-normal distribution as an approximation to the distribution of the sum of $n$ i.i.d. random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to zero at a rate of $n^{-2/3}$. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and $n$ is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.