Specifically, I would like to construct a field of order $2^{2n}$ with elements being $2\times2$ matrices whose entries are elements of $\mathbb{F}$.
I know the complex numbers can be represented as $2\times2$ real matrices, and I was trying to do something similar. I think the key is to find an expression $f(a, b)$ with $a, b\in\mathbb{F}$ such that $f(a, b)=0$ if and only if $a=b=0$. Then make sure the expression of the determinant is exactly $f(a, b)$ to get $p^{2n}-1$ invertible $2\times2$ matrices. I haven't had any luck with this idea though.
Any suggestions would be appreciated. I don't really want a specific answer since I want to do the thinking, but I'm hoping someone could guide me towards the right direction.