Approximate joint probabilities using marginal probabilities

Let $\displaystyle \mathbf Y=(Y_1,...,Y_n)^T$ be an n-dimensional Gaussian random vector with known mean vector and covariance matrix. I am interested in a joint probability that the elements are contained in some intervals, i.e. $\displaystyle P(Y_1 \in A_1,...,Y_n \in A_n)$.

In my particular case I have 2-dimensional marginal probabilities that take pairwise correlations into account easily available, i.e. $\displaystyle P(Y_1 \in A_1,Y_2 \in A_2), P(Y_2 \in A_2,Y_3 \in A_3), P(Y_1 \in A_1,Y_3 \in A_3)$ and so on.

Is there a nice way to (roughly) approximate the n-dimensional joint probability using several (or all) of the 2D-marginals?

TIA :)