Is tangent space a separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane?

Entire question should be:

Can we consider tangent space as the separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane? Is possible to view between plane (n+1) and hyperplane (n-1) a surface or a 'space between planes' like a subspace ?

Just hyperplane is a subspace so I need to try to understand if between plane-hyperplane we have a topology plane morphism or other algebraic structure/algebraic operation (spetre of a ring for example)