Approximate joint probabilities using marginal probabilities

Let 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. .

In my particular case I have 2-dimensional marginal probabilities that take pairwise correlations into account easily available, i.e. 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 :)