Basis of the space of polynomial functions with degree at most n over a finite field

Question

What is the dimension of the space of all polynomial functions of degree at most $n$ over the finite field $\mathbb{F}_2$?

I know that for over the field $\mathbb{R}$, the dimension is $n+1$, since we have the basis $1, x, x^2, \dots, x^n$. But I'm wondering how this would change if the field is $\mathbb{F}_2$ instead of $\mathbb{R}$.