topology on $\text{Hom}_{\mathbb R}(E,E')$ for two real vector bundles $E \to M$, $E'\to M$

Question

Let $\pi: E \to M$ and $\pi': E' \to M$ be two real vector bundles and denote the set of all linear maps between the fibres $E_p$ and $E'_p$ by $\text{Hom}_{\mathbb R}(E_p,E'_p)$. I want to show that the disjoint union$$\text{Hom}_{\mathbb R}(E,E') := \bigsqcup_{p \in M} \text{Hom}_{\mathbb R}(E_p,E'_p)$$ is a real vector bundle itself. Let's denote the elements of $\text{Hom}_{\mathbb R}(E,E')$ as pairs $(p,L)$ where $p \in M$ and $L \in \text{Hom}_{\mathbb R}(E_p,E'_p)$.
An obvious candidate is the surjection $\sigma: \text{Hom}_{\mathbb R}(E,E') \to M,\ \ (p,L) \mapsto p$. So we are left to show that $\sigma$ is continuous and that there are local trivialisations.

My problem is that I don't know what the topology on $\text{Hom}_{\mathbb R}(E,E')$ is and that I don't know any book which covers this example.