Can the tangent bundle $TM$ be considered as the product manifold of $M$ and its tangent planes?

Question

Can the tangent bundle $TM$ be considered as the product manifold of a smooth manifold $M$ and its tangent planes $T_pM$ with $p \in M$?.
If this is the case, then can we conclude that it is a smooth manifold because it is a product of smooth manifolds and we can find a differentiable structure that comes from the maps of $M$ and the maps of $T_pM$?