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, and here you can find obstructions and a nice history of the problem, as well as many references.

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 gives obstructions and a nice history of the problem, as well as many references.