# Smooth structure and vector bundle structure on $L_{alt}^k(TM)$

I want to exhibit smooth structure and vector bundle structure on $$L_{alt}^k(TM)=\bigcup_{p\in M} L_{alt}^k(T_pM)$$ where $$M$$ is a manifold of dimension $$n$$ and $$L_{alt}^k(T_pM)$$ is the set of all $$k$$-th alternating multilinear maps on $$T_pM$$.

I can guess that for a chart $$(U,x)$$ of $$M$$ , I have to construct some map form $$\bigcup_{p\in U} L_{alt}^k(T_pM)$$ to $$U×\Bbb R^{\binom{n}{k}}$$.I don't know deep theory of vector bundle, so some rigorous proof or reference that contains all explanation on this topic, will be helpful to me.