vector spaces - Explicit example of non constant linear functional $f: mathbb R to mathbb Q$?

Recall that $V=\mathbb R$ is a uncountably dimension vector space over $\mathbb Q$ as countable dimension vector space over $\mathbb Q$ is itself countable.

Is there any explicit example of a non constant linear functional $f: \mathbb R \to \mathbb Q$ ?

Existence of such linear functional is almost trivial but can we give the explicit example of such $1$-form? Also it is clear that under usual topology such a map $f$ cannot be continuous as $\mathbb Q$ is totally disconnected.