The second Stiefel-Whitney class $$w_2(TM) \in H^2 (M;\mathbb{Z}_2)$$ measures the obstruction to finding a 3-frame over the 2-skeleton. If $w_1$ was trivial, the manifold being oriented, $T$M can already be trivialized over the 1-skeleton. And we picked an orientation of $M$, then by using this orientation we can complete any 3-frame to a 4-frame. Therefore we can say that, for oriented manifolds, $w_2(TM)$ is the obstruction to trivializing $TM$ over the 2-skeleton31 of $M$.
This is what described in Alexandru Scorpan: The Wild World of 4 Manifold p.162:
Question: so how do we write the second Stiefel-Whitney class $w_2(TM)$ as a cochain field explicitly (please be as explicit as possible for your construction)?