The Steenrod squares $Sq^i: H^n(-;\mathbb{F}_2) \to H^{n+i}(-;\mathbb{F}_2)$ are fundamental cohomological operations. By Yoneda lemma, it induces a map between the Eilenberg-MacLane spaces $K(\mathbb{F}_2; n) \to K(\mathbb{F}_2; n +i)$. By Dold-Kan correspondence, this map should be expressible as a chain map (if I'm not mistaken): $$\widehat{Sq^i}: \mathbb{F}_2[-n] \to \mathbb{F}_2[-(n+i)].$$

Question: How to explicitly describe $\widehat{Sq^i}$? Is it much harder to do this for other finite fields $\mathbb{F}_q$?