You are likely looking for the vectorized version which is called multiple (linear) regression:
$y_i=\beta_0+\vec{x}_i~\vec{\beta}=\beta_0+\beta_1 x_{1,i}+\beta_2 x_{2,i}+\dots+\beta_nx_{n,i}+\varepsilon_i$. Where the $\varepsilon_i$-s are assumed iid and $\varepsilon_i\sim N(0,\sigma^2)$
It's pretty much the same as univariate linear regression, except you're fitting a plane and not a line.
The $\vec\beta$ is estimated using the augmented notation: $\vec{b}=\hat{\vec\beta}=(X^{T}X)^{-1}X^{T}Y$ where $X$ is a matrix with all your $x_i$-values, and augmented with 1s. Read more here under "Least squares estimations"