On the spacetime manifold in general relativity, one chooses a basis at a point and express it by the partial derivatives with respect to the four coordinates in the coordinate system. And then the basis vectors in the dual space will be the differentials of the coordinates. Why do one do that? I understand that by doing so it allows one to identify the vectorfields on the manifold by a Lie algebra. But why do one choose to do so? And why is it so important for allowing a Lie algebra of the vector fields to be defined? Could someone give a more intuitive explanation for this?