Variance Among Coordinates of Multivariate Normal

Sample $(X_1, X_2,\ldots, X_n)^T\sim{N(\textbf{0}, \Sigma)}$ . What is the expected cross-sectional variance of $(X_1, X_2, \ldots, X_n)^T$ ? In other words, if $$ S^2 = \frac{1}{n}\sum_{k = 1}^n \left(X_k - \bar{X}\right)^2\qquad\text{and}\qquad \bar{X} = \frac{1}{n}\sum_{k = 1}^n X_k, $$ what is $E[S^2]$ ? As an example, I'll show the relatively trivial two dimensional case. Suppose $$(X_1, X_2)^T\sim{N\left(\textbf{0}, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}\right)}. $$ Then the cross-sectional mean is $$ \bar{X} = \frac{X_1+X_2}{2}. $$ Using the cross-sectional mean, the cross-sectional variance is $$ S^2 = \frac{1}{2}\left(X_1 - \bar{X}\right)^2 + \frac{1}{2}\left(X_2 - \bar{X}\right)^2 = \left(\frac{X_1 - X_2}{2}\right)^2 $$ Hence, the expected cross-sectional variance is $$ E\left[S^2\right] = E\left[\left(\frac{X_1 - X_2}{2}\right)^2\right] = \frac{\sigma_1^2 - 2\rho\sigma_1\sigma_2+\sigma_2^2}{4}. $$