I attempt to understand the coordinate representation of multilinear functions from An Introduction to Manifolds (Second Edition) by Loring Tu. Below I quote the relevant part of the book (page no. 31).
3.10 A Basis for $k$-Covectors
Let $e_1, \cdots, e_n$ be a basis for areal vector space V, and $\alpha^1, \cdots, \alpha^n$ be the daul basis for $V^\vee$. Introduce the multi-index notation $$I = (i_1, \cdots, i_k)$$ and write $e_I$ for $(e_{i_1}, \cdots, e_{i_k})$ and $\alpha^I$ for $\alpha^{i_1} \wedge \cdots \wedge \alpha^{i_k}$.
A $k$-linear function $f$ on $V$ is comepletely 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 $I$ in strictly ascending order.
My Question
I don't see how $f$ being alternative is completely determined by its values on $(e_{i_1}, \cdots, e_{i_k})$ with $1\leq i_1 < \cdots <i_k \leq n$. Can you please explain it to me, if possible by an example?
My Attempts
At first, I consider a multilinear function $f: V^k \to \mathbb{R}$ so that
\begin{eqnarray} f(v_1, \cdots, v_k) &=& f\left(\sum_{i_1=1}^{n} v_{\color{red}{1}}^{i_1} \, e^{\color{red}{1}}_{i_1}, \cdots \cdots, \sum_{i_k=1}^{n} v_{\color{red}{k}}^{i_k} \, e^{\color{red}{k}}_{i_k}\right) \\ &=& \sum_{i_1=1}^{n} \cdots \sum_{i_k=1}^{n} v_{\color{red}{1}}^{i_1} \cdots v_{\color{red}{k}}^{i_k} f\left(e^{\color{red}{1}}_{i_1}, \cdots \cdots, e^{\color{red}{k}}_{i_k}\right), \end{eqnarray} and therefore is comepletely determined by its values on all $k$-tuples $(e_{i_1}, \cdots, e_{i_k})$.
But I can't derive the case for the alternating $k$-linear function from here.