No, this is not the covariance matrix but itsinverse. If it was the covariance matrix, you could read the variance from the diagonal coefficients...

The following way works: integrate the marginals one after another (from both sides toward the middle variable ; by symmetry you just have to do one side), do it by hand for the first ones in order to get a pattern suitable for a proof by induction. This way you can resume to studying a real-valued sequence defined by induction (something like where I don't specify what I'm dealing with...), and you need an asymptotic expansion for this sequence. This becomes a calculus question, where various methods apply (you should be able to even get an asymptotic equivalence for the variance).

I let you try to perform this computation; tell me if you don't succeed.