linear algebra – can you describe the span of a two dimensional vector space, as the solutions of a degree 1 polynomial in the affine field $K^n$?

I will just like to prove this statement to help me understand affinge geometry better.

can you describe the span of a two dimensional vector space, as the solutions of a degree 1 polynomial in the affine field $K^n$?

E.g
For $mathbb{R}^3$, any two dimension subspace is a hypersurface, hence can be described as,
$ax+by+cz+d=0$,
I was wondering if this could be generalised, and if you can provdie a proof or literature that could help.