If you know it, use inflation/restriction to a 2-Sylow subgroup $S \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ of $A_4$, you will find that $H^2(S,\mathbb{Z}/2\mathbb{Z}) \simeq H^2(A_4,\mathbb{Z}/2\mathbb{Z})$ (and this isomorphism is explicit), and $H^2(S,\mathbb{Z}/2\mathbb{Z})$ has eight elements: one corresponding to the direct product $S \times \mathbb{Z}/2\mathbb{Z}$, three to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, three to the dihedral group having eight elements, and one to the quaternion $8$-group $D$ ($D/ \{\pm 1 \} \simeq S$). You are interested in the latter (that's the preimage of $S$). It is easy to write this cocyle explicitly. But at that point it is easier to define the binary tetrahedral group as the Hurwitz quaternions units, pick a section, and compute.
Otherwise just take a big piece of paper, and fill an array with $0$s and $1$s.