A map $f: M \to \mathbb{R}^n$ is said to be $k$-regular if whenever $x_1, \dots, x_k$ are distinct points of $M$, then $f(x_1), \dots, f(x_k)$ are independent.  There is an abundance of literature on $k$-regular maps.  [Blagojević, Lück, and Ziegler - On highly regular embeddings](http://arxiv.org/abs/1305.7483) gives obstructions and a nice history of the problem, as well as many references.