Consider a smooth manifold $\mathcal{M}$ of dimension $d$ and some smooth $\mathbb{R}$-vector bundle $E$ of rank $d$. On wikipedia, it is claimed that a bundle-morphisms of the type $T\mathcal{M}\to E$ are one and the same as $E$-valued $1$-forms, i.e. there is an isomorphism
$$\Omega^{1}(\mathcal{M},E)\cong\mathrm{Hom}(T\mathcal{M},E).$$
However, I cannot see how this isomorphism is defined explicitely. I know that there is a $C^{\infty}(\mathcal{M})$-module isomorphism $\Omega^{1}(\mathcal{M},E)\cong \mathrm{Hom}(\mathfrak{X}(\mathcal{M},\Gamma^{\infty}(E))$, but I do not know if this helps.
Furthermore, is it true that the set of bundle-isomorphisms is in one-to-one correspondence with the non-degenerate 1-forms, i.e.
$$\Omega^{1}_{\mathrm{nd}}(\mathcal{M},E)\cong\mathrm{Iso}(T\mathcal{M},E)?$$
I think I have read this somewhere, but I can't remember where exactly.