Let $\{U_\alpha\}$ be open cover of space $B$ and let $g_{\beta\alpha}:U_\alpha\cap U_\beta\to GL_n(\mathbb R)$ be family of maps that satisfy cocycle condition: $g_{\gamma\beta}g_{\beta\alpha}=g_{\gamma\alpha}$ on $U_\alpha\cap U_\beta\cap U_\gamma$.
Further, let $E$ be quotient of disjoint union $\sqcup_\alpha(U_\alpha\times\mathbb R^n)$ obtained by identifying $(x,v)\in U_\alpha\times\mathbb R^n$ with $(x,g_{\beta\alpha}(x)(v))$.
I want to know how $E$ is vector bundle over $B$, more precisely, what are local trivializations of $E$?