Let (X 1 , X 2 ,. .. , X n) be a random vector and denote by P (X1,X2,...,Xn) its probability distribution on R n. Inspired by [11], we develop a multidimensional Stein-Malliavin calculus which allows to measure the Wasserstein distance between the law P (X1,X2,...,Xn) and the probability distribution P Z ⊗P (X2,...,Xn) , where Z is a Gaussian random variable. That is, we give estimates, in terms of the Malliavin operators, for the distance between the law of the random vector (X 1 , ..., X n) and the law of the vector (Z, X 2 , ..., X n), where Z is Gaussian and independent of (X 2 , ..., X n). Then we focus on the particular case of random vectors in Wiener chaos and we give an asymptotic version of this result. In this case, if we consider a n-dimensional sequence with components in Wiener chaos, such that the first component is asymptotically Gaussian and the rest of the vector converges to an arbitrary limit, we are able to evaluate the distance between this vector and its limit. To illustrate our method, we derive the rate of convergence for the Wasserstein distance for a two-dimensional sequence of multiple stochastic integrals, the first converging to a normal law and the second to a Rosenblatt distribution.