I suspect that this is a very simple question, but I need to ask.

My question is

How do the fields of characteristic $p$ look like?

If $K$ is a finite field of order $p^n$, then $K$ has characteristic $p$ ($p$ prime). We can take the algebraic closure of $K$ and we get $$ \bar{K} = \bigcup_n K^n. $$ Then $K$ has characteristic $p$ as well.

Are all the algebraically closed (hence infinite) fields of characteristic $p$ algebraic closures of a union of finite fields in this way?