Let $V$ be a real $d$-dimensional vector space, and let $1<k<d$ be fixed. Let $v_i$ be a basis for $V$. Consider the induced basis for the $k$-th exterior power $\bigwedge^k V$, given by $v_{i_1} \wedge \dots \wedge v_{i_k}$.
Suppose we have another basis $w_i$ for $V$, such that every $v_{i_1} \wedge \dots \wedge v_{i_k}$ equals to some $\lambda_{i_1,\ldots,i_k}w_{j_1} \wedge \dots \wedge w_{j_k}$ where $\lambda_{i_1,\ldots,i_k} \in \mathbb R$ are non-zero scalars. Is it true that after a possible rearrangement of the $w_i$, we must have $\text{span}(v_i)=\text{span}(w_i)$?
i.e. does there exist a permutation $\sigma \in S_d$ such that $\text{span}(v_i)=\text{span}(w_{\sigma(i)})$?
This question can be reformulated as a combinatorial question as follows:
By the independence of the $v_{i_1} \wedge \dots \wedge v_{i_k}$, for each strictly increasing multi-index $I=(i_1,\ldots,i_k)$, there corresponds a unique multi-index $J=(j_1,\ldots,j_k)$ such that
$v^I:=v_{i_1} \wedge \dots \wedge v_{i_k}=\lambda_Iw_{j_1} \wedge \dots \wedge w_{j_k}=\lambda_Iw^J$ holds. In other words, there exist a permutation $\tau \in S_{\binom{d}{k}}$ such that for every multi-index $I$, $v^I=\lambda_I w^{\tau(I)}$.
The question is whether or not $\tau$ is induced by a permutation $\sigma \in S_d$ in the obvious way: $\tau((i_1,\ldots,i_k))=(\sigma(i_1),\ldots,\sigma(i_k))$.
Indeed, if there exist such a permutation $\sigma$, then by setting $u_i:=w_{\sigma(i)}$, we are in the following situation:
$v_{i_1} \wedge \dots \wedge v_{i_k}=\lambda_Iu_{i_1} \wedge \dots \wedge u_{i_k}$, which implies $$\text{span}(v_{i_1},\dots,v_{i_{k-1}},v_{i_k})=\text{span}(u_{i_1},\dots,u_{i_{k-1}},u_{i_k}). \tag{1}$$
By switching between $i_k$ and $i_{k+1}$ in $(1)$,we obtain
$$\text{span}(v_{i_1},\dots,v_{i_{k-1}},v_{i_{k+1}})=\text{span}(u_{i_1},\dots,u_{i_{k-1}},u_{i_{k+1}}). \tag{2}$$
By intersecting (1) and (2), we deduce that
$$\text{span}(v_{i_1},\dots,v_{i_{k-1}})=\text{span}(u_{i_1},\dots,u_{i_{k-1}}). \tag{3}$$
In the passage from $(1)$ to $(3)$ we have "removed" the last vectors $v_{i_k},u_{i_k}$. Continuing in this way, we can remove all vectors, until we get to $\text{span}(v_{i_1})=\text{span}(u_{i_1})$ as required.
Best Answer
Yes, you can see it from basic geometry and combinatorics. Given a subset $I \subseteq [d]$, set $P_I = \operatorname{span} \{ v_i \}_{i \in I}$. Note that $P_I \neq P_J$ if $I \neq J$ and we have the nice property that $P_{I \cap J} = P_I \cap P_J$. Now, let's fix $1 \leq k < d$ and assume we are given the collections of subspaces $\{ P_I \}_{|I| = k}$ as a set (that is, without a specific order). Can we reconstruct the vectors $v_i$ up to permutation and scaling? Let's choose $n - 1 \choose k - 1$ subspaces which correspond to some collection of subsets $\mathcal{F}$ of size $k$ of $[n]$ and intersect them to get
$$ \bigcap_{I \in \mathcal{F}} P_I = P_{\bigcap_{I \in \mathcal{F}} I} $$.
We have only two options:
The intersection $\bigcap_{I \in \mathcal{F}} I$ cannot contain two or more elements because the number of subsets which contain two specific elements is ${n - 2 \choose k - 2} < {n - 1 \choose k - 1}$. In addition, by running over all choices of subsets $\mathcal{F}$ of size ${n - 1 \choose k - 1}$ we will get all the subspaces $P_{\{ i \}}$ for $1 \leq i \leq d$.
Returning to your question, let's set $Q_I = \operatorname{span} \{ w_i \}_{i \in I}$. According to your assumption, the collections $\{ Q_I \}_{|I| = k}$ and $\{ P_I \}_{|I| = k}$ are identical and then by what I wrote above we must have that the collections $\{ Q_I \}_{|I| = 1}$ and $\{ P_I \}_{|I| = 1}$ are identical which implies that the $v_i$'s and the $w_i$'s are identical up to permutation and scaling.