When we study Differential Geometry of surfaces, we define something called the first fundamental form. It is a symmetric bilinear form on the tangent space at a point on the surface.
Moreover, I know from Linear Algebra that the determinant of this first fundamental form (when we choose a parametrisation for our surface, and represent this bilinear form in a matrix form) is exactly the Gram determinant, which is equal to the square of the volume of a parallelepiped in this tangent space.
Seeing this connection, and also considering the fact that I know more about smooth manifolds compare to my knowledge on surfaces, is there any generalisation of the concept of first fundamental form to smooth manifolds ?