How to obtain parameters for conditional gaussian for more than two variables?

Question

We can obtain $\mu$ and $\Sigma$ for a conditional gaussian with two variables, $a$ and $b$, $p(x_{a}|x_{b})$ like this:

$$\mu_{a|b}= \mu_{a}+\Sigma_{ab}\Sigma^{-1}_{bb}(x_{b}-\mu_{b})$$ $$\Sigma_{a|b} = \Sigma_{aa}-\Sigma_{ab}\Sigma^{-1}_{bb}\Sigma_{ba}$$

Question: How do I extend obtaining $\mu$ and $\Sigma$ for 3, 4, ..., $n$ conditioning variables?

Reference: Bishop-Pattern-Recognition-and-Machine-Learning-2006