Let $V$ be a vector space over $K$ with basis $\{e_1,\ldots,e_n\}$. I already showed that for a fixed tuple $I = (i_1,\ldots,i_r)$ with $1 \le i_1 < i_2 < \cdots < i_r \le n$, there exists a multilinear map $\varphi_I: V^r \rightarrow K$ such that
\begin{align} \varphi_I(e_{j_1}, \ldots, e_{j_r}) &= \begin{cases} {sgn(\sigma)} \quad \text{if there exists a permutation $\sigma$ of $[r]$ with $j_{\sigma(k)} = i_k$ for all $k \in [r]$}\\ 0 \qquad \text{else}\end{cases} \end{align}
for all $(j_1,\ldots,j_r) \in [n]^r$.
Show that $\varphi_I$ is alternating.
I would have said that $\varphi_I$ is alternating simply because if there is a permutation $\sigma$ of $[r]$ such that $\varphi_I(e_{j_1}, \ldots, e_{j_r}) = sgn(\sigma)$, then, as we know from linear algebra, transposing $e_{j_k}$ and $e_{j_l}$ just changes the sign of $\sigma$. (We may assume that $e_{j_1}, \ldots, e_{j_r}$ are distinct, otherwise $\varphi_I = 0$ and the claim is clearly true then.) However, my instructor said that there is some trap in this exercise, so is there something I am overlooking?