Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear subspace of $\mathbb R^n$, i.e. for every such subspace $H \subset \mathbb R^n$ it holds that $$ \gamma([0,1]) \cap (\mathbb R^n \setminus H) \neq \emptyset. $$ Is there any theory in this direction?

Any help would be highly appreciated!