riemannian geometry – Differential of a canonical map

While studying about curvatures I came up with the following but was unable to work it out fully. Let $M subset mathbb{R^n}$ be an embedded submanifold of dimension $k$. Then there is a natural (smooth?) map
$$alpha: M rightarrow text{Gr}(k,mathbb{R}^n),$$

given by $p mapsto T_p M$. I know the grassmanian admits a natural description of its tangent spaces, namely:
$$T_Vtext{Gr}(k,mathbb{R}^n) cong text{Hom}(V , V^+)$$

where by $V^+$ I mean the orthogonal complement. Is there a nice description of this map? I couldn’t work it out. Also, a closesly related question: suppose we look instead of at $p mapsto T_pM$ at $p mapsto (T_p M)^+$, in the codimension 1 case, I think the differential + a choice of normal vector field gives you a bilinear form on $T_pM$ automatically. Intuitively I would expect this is something like the second fundamental form of the manifold , but again, when trying to work it out it becomes a mess. Any ideas? Thanks!

P.S. If someone knows a universal property the grassmanians satisfy, I’m also very interested!