I am studying differential $k$-forms from Tu's Introduction to Manifolds. If $(U, x^1, \ldots,x^n)$ is a coordinate chard for a manifold, then the dual basis for the cotangent space is given by $$(dx^1)_p, \ldots, (dx^n)_p.$$ He states that by an earlier proposition, the basis for the alternating $k$-tensors in $\Lambda^k(T^*_pU)$ is given by $$(dx^{i_1})_p \wedge\cdots\wedge(dx^{i_k})_p, \quad 1 \leq i_1 < \cdots < i_k \leq n.$$ In the section where he introduces the mentioned proposition (where he is only focusing on $\mathbb{R}^n$), he states that
A $k$-linear function $f$ on a real vector space $V$ is completely determined by its values on all $k$-tuples $(e_{i_1}, \cdots, e_{i_k})$. If $f$ is alternating, then it is completely determined by its values on $(e_{i_1}, \cdots, e_{i_k})$ with $1 \leq i_1 < \cdots < i_k \leq n$; that is, it suffices to consider $e_I$ with $$I = (i_1, \ldots, i_k)$$ in strictly ascender order.
I was could not figure out why the above suffices.