Proving pairwise independence is equivalent to uncorrelated [duplicate]

The problem states: "Let $X = (X_1, X_2, \dots, X_n)^T$ be a multivariate normal random vector. Show that $X_i$ and $X_j$, $i \neq j$, are independent if and only if $X_i$ and $X_j$ are uncorrelated."
Going from pairwise independence to uncorrelated is easy: Just use the fact that $E(X_i X_j) = E(X_i)E(X_j)$ to show that $\mathrm{Corr}(X_i X_j)=0$. But how to prove the converse? Uncorrelated doesn't imply independence, does it?
Hint: Write the joint probability density function for $X_1, \dots, X_n$ given that their correlation matrix is diagonal (they are pairwise uncorrelated). Then try to write the probability density function as a product of the probability density functions for $X_1$ and that of $X_2$, and so on. If you can factor the probability density function, then they are independent.