I am trying to prove the following:

Let $(X\_1,\dots,X\_n)$ be a Gaussian vector with mean 0 and covariance matrix B. Find the distribution of $E(X\_1\mid X\_2,\dots,X\_n).$

I know in general for two Gaussian r.v. $X\_1$ and $X\_2$ we can show that $f\_{X\_1|X\_2} (x\_1|x\_2) = \frac{1}{\sigma\_{X\_1} \sqrt{2 \pi (1-\rho^2)}}\exp\frac{(-(x\_{1}- \rho(\sigma\_{X\_1}/ \sigma\_{X\_2})x\_2)^2)}{2 \sigma^{2}\_{X\_1}(1- \rho^{2})}$ where $\rho$ is the correlation and

$E(X\_{1}|X\_{2})=\int x\_{1} f\_{X\_1|X\_2} (x\_1|x\_2)dx\_{1}. $

How can I generalize this for a gaussian random vector of size $n$? Can I conclude on the distribution of $E(X\_1|X\_2,...X\_n)$ based on the form of $f\_{X\_1|X\_2,\dots,X\_n} (x\_1|x\_2,\dots,x\_n)$? For example, in the case of 2 Gaussian r.v we produce a normal r.v with variance $2 \sigma^{2}\_{X\_1}(1- \rho^{2})$ and mean $\rho(\sigma\_{X\_1}/ \sigma\_{X\_2})x\_2$.