Why the combinatorial second Stiefel-Whitney class is a cocycle?

Question

From the book "Spin geometry" by Lawson&Michaelson Appendix A or this literature we know that there is a nice combinatorial way to interpret the second SW class by the transition functions of a principal bundle. The basic idea is, we firstly take a good covering, where the principal bundle is trivial on each open sets. Then on each two-fold intersection we have the transition functions $$g_{\alpha\beta}: U_\alpha\bigcap U_\beta\rightarrow \mathrm{SO}(n)$$ Lifting each map to $\overline{g}_{\alpha\beta}:U_\alpha\bigcap U_{\beta}\rightarrow \mathrm{Spin}(n)$ gives a cochain on each three-fold intersection $$w_{\alpha\beta\gamma}=\overline{g}_{\alpha\beta}\overline{g}_{\beta\gamma}\overline{g}_{\gamma\alpha}:U_{\alpha}\bigcap U_{\beta} \bigcap U_{\gamma}\rightarrow \mathbb{Z}_2$$ On both references above they just claim that $w$ is a cocycle. But I just wonder is the argument really so trivial?