Skew-symmetricity of the Cup product

Question

Let $f, g \in H^1(G, {\Bbb Z}/n{\Bbb Z})$ be two elements. Consider the cup-product

\begin{equation} ∪ \colon H^1(G, {\Bbb Z}/n{\Bbb Z}) \times H^{1}(G, {\Bbb Z}/n{\Bbb Z}) \to H^2(G, {\Bbb Z}/n{\Bbb Z}). \end{equation} This is defined as follows:

$f \cup g(\sigma,\tau) \colon= f(\sigma)\sigma (g(\tau)) \in {\Bbb Z}$.

Q. How can one show that the cup-product $∪$ is skew-symmetric explicitly? That is, $f \cup g = -g \cup f$.