Is it always possible to order every vector of $\mathbb{F}^n_2$ (except the zero vector) as a sequence $V = (v_1, \dots, v_{2^n - 1}) | v_i \in \mathbb{F}^n_2 \setminus \{0\} $ such that every subsequence of $V$ of length $n$ is a basis of $\mathbb{F}^n_2$? Furthermore, is it always possible to apply this rule (subsequences of length $n$ make a basis for $\mathbb{F}^n_2$) to concatenations of such a sequence (so that the rule "wraps around")?
If so, what's a good way to construct such a sequence for a given $n$?
Here's such an example sequence for $\mathbb{F}^3_2$ that can also "wrap around": $$v_1 = (0, 0, 1)$$ $$v_2 = (0, 1, 0)$$ $$v_3 = (1, 0, 0)$$ $$v_4 = (0, 1, 1)$$ $$v_5 = (1, 1, 0)$$ $$v_6 = (1, 1, 1)$$ $$v_7 = (1, 0, 1)$$