Can span of two vectors in $\mathbb{R}^3$ generate whole $\mathbb{R}^2$ plane?

Question

Suppose I have two vectors in $\mathbb{R}^3$ and I linearly combine them. Can and will they generate whole $2\text D$ plane? My teacher did an example and showed that such vectors can't span $\mathbb{R}^2$ because they will be a $2\text D$ plane (a sheet) in $3\text D$ ($\mathbb{R}^3$). Kindly answer with example.

Answer 1

Certainly $(1,0,0)$ and $(0,1,0)$ span a sheet, which, as in the case of all sheets, is isomorphic to $\mathbb R^2$.

But no two vectors can span all sheets in $\mathbb R^{\color{blue}3}$, as then we would have two vectors spanning a three dimensional space, which is impossible...