Is this a valid definition of an orientable manifold?

Question

That is, an n-dimensional manifold M is said to be orientable iff the tangent spaces of its points are consistently oriented(i.e. for any 2 distinct points p and q, the matrix $B_{pq}$ which transforms the basis vectors of $T_{p}M$ into the basis vectors of $T_{q}M$ has a positive determinant). In particular, is this equivalent to the definition of manifold orientability using homology groups?