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: $$T_V\text{Gr}(k,\mathbb{R}^n) \cong \text{Hom}(V , V^+)$$
where by $V^+$ I mean the orthogonal complement. Is there a nice description of its differential at an arbitrary point $p$? 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!