# Variance of Residuals in Multiple Linear Regression

If I have $$n$$ variables $$x_1,\dots,x_d$$ in my dataset, and I regress the first one against all the others i.e.

$$x_1=a_{12}x_2 + a_{13}x_3 + \dots + a_{1d} x_d + \epsilon_1$$,

how can I calculate $$\text{Var}(\epsilon_1)$$?

If there are $$n$$ datapoints for each variable, then it is possible to calculate $$\Sigma_{2:d}$$ i.e. the covariance matrix for variables $$x_2,x_3,\dots,x_d$$. I can also easily calculate an unbiased estimate for $$\text{Var}(x_1) = \frac{1}{n-1} \displaystyle\sum_{i=1}^n(x_1^i - \bar{x}_1)^2$$.

But I don't know what to do with these?