Can we find $n^2$ fullrank matrices in $\mathbb{F}^{n \times n}$ which are linearly independent (i.e. when vectorized are linearly independent)? If not, how many such matrices can be found?

If the characteristic of $\mathbb{F}$ is not two and $E_{i,j}$ $(1\le i,j\le n)$ is the "standard basis", then the matrices $I+E_{i,j}$ are invertible. They are linearly independent if $n+1\ne0$ in $\mathbb{F}$. If the characteristic of $\mathbb{F}$ is greater than 2, we can use $IE_{i,j}$ instead and these are linearly independent if $n1\ne0$. So we have explicit examples except in characteristic two. 

