I found in a physics paper these equations for Group cohomology of $\mathbb{Z}_n$:
$$ H^d\big[ \mathbb{Z}_n\, , \,U(1)\, \big] = \left\{ \begin{array}{cl} U(1) & d=0 \\ \mathbb{Z}_n & d \equiv 0 \pmod 2\\ \mathbb{Z}_1 & d \equiv 1 \pmod 2 \end{array} \right. $$
In a sense, the Group cohomology is nothing more than a set of equivalence classes of functions $\mathbb{Z}_n \to U(1) = \{ e^{2\pi i \, x} : x \in \mathbb{R} \} \simeq \mathbb{R} / \mathbb{Z}$.
Can we write down explicit representatives for $H^2[\mathbb{Z}/n \mathbb{Z}]$ or $H^3[\mathbb{Z}/n \mathbb{Z}]$ as functions ? Is $e^{2\pi i n \, x^2} \in H^2$ ?