I was thinking about if $\mathbb R$ could be written as $(-\infty , \infty)$. I'm not sure if it's okay, because I've read somewhere (I can't remember where) that $(-\infty , \infty)$ declares extended real line, which is totally different from $\mathbb R$.
It can be written that way, it doesn't mean it should be.
Have you ever heard of "complex infinity" or "directionless infinity"? Your readers would quickly realize you're not referring to either of those, but it could still be enough to cause a little cognitive dissonance and disrupt the flow of your presentation.
If you don't like $\mathbb R$ for whatever reason, you can always use $\textbf R$ instead. You don't lose any clarity that way.