Extended Cramér-Rao inequalities of estimation theory, together with related Cramér-Rao inequalities saturated by generalized $q$-Gaussian distributions, were presented in a recent paper. These results introduced an extended version of Fisher information and a new characterization of generalized $q$-Gaussian distributions which are important in several areas of physics and mathematics. In the present work, we extend these results to the mutidimensional case. We show how the generalized Cramér-Rao inequality for the estimation of a parameter can be extended to the mutidimensional case, with a formulation that involves a general norm on $\mathbb{R}^{n}$ and its dual norm. As a particular case, we obtain a new multidimensional Cramér-Rao inequality which is saturated by generalized $q$-Gaussian distributions. We give another related Cramér-Rao inequality, for a general norm, which is also saturated by these distributions. These results yield a new information theoretic characterization of generalized $q$-Gaussian distributions