There is a bunch of such statements which can be obtained by [Stein's method][1].

You might be interested in the paper ["On the Rate of Convergence in the Multivariate CLT"][2] 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*][3] by Chen, Goldstein and Shao.



  [1]: https://en.wikipedia.org/wiki/Stein%27s_method
  [2]: https://projecteuclid.org/euclid.aop/1176990448
  [3]: https://books.google.co.uk/books?id=5jMVpAbs9UkC&printsec=frontcover&dq=Normal+Approximation+by+Stein%2527s+Method&hl=en&ei=UFbsTPL-FcShOqrIzJEB&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false