orientations of a manifold and its tangent bundle

Question

Milnor's characteristic classes: Suppose M is an orientable manifold. An orientation for M is a function which assigns to each point x of M a preferred generator $u_x$ for the infinite cyclic group $ H_n(M,M-x)$, using integer coefficients. These preferred generators are required to "vary continuously" with x, in the sense that $u_x$ corresponds to $u_y$ under the isomorphisms $$ H_n(M,M-x) \leftarrow H_n(M,M-N) \to H_n(M,M-y)$$where N denotes a nicely embedded n-cell neighborhood of x and y is any point of N. similarly, an orientation for the vector bundle $r_M$ can be specified by assigning a preferred generator $v_x$ to the infinite cyclic group $H_n(DM_x,DM_x-0)$ for each x. These generators also vary continuously in the sense: $$H_n(DM_x,DM_x-0) \to H_n(DN,DN-(N\times0)) \leftarrow H_n(DM_y,DM_y-0)$$ it is said that the homology group $H_n(M,M-x)$ is canonically isomorphic to $H_n(DM_x,DM_x-0)$ by viewing x as a 0-dimensional manifold, embedded in M as a closed subset with normal bundle $DM_x$. So the two kinds of generators above can be related. But I have no idea how to prove that $u_x$ varies continuously with x if and only if the corresponding generators $v_x$ vary continuously with x under this relation. So how to prove it? In other words, how to prove that M is orientable if and only if its tangent bundle is orientable using the homology language as above?