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For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:

Are similar statements known for the multivariate version of the central limit theorem, that use some standard distance measure?

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There is a bunch of such statements which can be obtained by Stein's method.

You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically devoted to Berry-Esseen theorems in the multidimensional setting. Have a look also at the very recent book Normal Approximation by Stein's Method by Chen, Goldstein and Shao.

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