Proof of existence of Schauder basis for $L^p(\Omega)$?

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like
All classical (separable) Banach spaces used in analysis have a Schauder
basis .
I was wondering where to find a proof confirming this statement for
$L^p(\Omega)$ with a domain $\Omega \subset \mathbb R^d$, $d\in\{2,3\}$,
and with $1<p<\infty$.
In the book Bases in Banach Spaces by Singer, where also Brèzis references
to, there is a proof for $L^p([0,1])$. Maybe, I just miss how to simply
extend the arguments to higher dimensions.