You are likely looking for the vectorized version which is called [multiple (linear) regression](https://newonlinecourses.science.psu.edu/stat501/node/311/):

$y_i=\beta_0+\mathbf{x_i}~\mathbf{b}=\beta_0+\beta_1 x_{1,i}+\beta_2 x_{2,i}+\dots+\beta_nx_{n,i}+\varepsilon_i$. 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.