Elements of a Module Whose Annihilator is a Given Associated Prime

Question

Given a commutative ring $R$ and an $R$-module $M$, an associated prime of $M$ is a prime ideal $\mathfrak{p} \subset R$ such that $\mathfrak{p} = \operatorname{Ann}(x)$ for some $x \in M$. I'm interested in the problem of finding all elements of $M$ whose annihilator is $\mathfrak{p}$. I was able to show that multiples $ax$, where $a \notin \mathfrak{p}$, have this property but I don't know how to get a complete list. Is there a general way to do it or would I have to rely on an ad hoc argument for the specific ring and module I'm working with?