One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of $H^1(SO(n);Z_2)$. Then this is equivalent to say that: (1) $\sigma$ determines a double cover of $E$, such that (2) each fiber $SO(n)$ is covered by its double cover $spin(n)$.
I do not understand how the definition implies (1).