Let $ M $ be a smooth manifold. Given $ p\in M $ denote with $$ \langle{-},{-}\rangle_p\colon \mathrm T_p^*M\otimes \mathrm T_pM\to \mathbb R $$ the canonical duality pairing between the tangent space $ \mathrm T_pM $ of $ M $ at $ p $ and its dual.
Can this map be "lifted" to a morphism of vector bundles over $ M $ $$ \langle{-},{-}\rangle\colon \mathrm T^*M\otimes \mathrm TM\to \mathbb R $$ such that $$ \langle \omega,X\rangle = \langle \omega,X\rangle_p $$ for every $ \omega\in \mathrm T_p^*M $ and $ X\in \mathrm T_pM $?
Could this construction be generalized to the case where we have two vector bundles $ (E,\pi,M) $ and $ (E^\prime,\pi^\prime,M) $ over $ M $ and a family of duality pairings $$ B_p\colon E_p\otimes E_p^\prime\to \mathbb R $$ one for each $ p\in M $?
I'm asking because I think I've checked that everything is working properly but I've never seen anyone talk about this construction, so I'm a bit suspicious that I might have done something wrong.