Exterior power of a space of maps $(\mathbb{K}^T)$

Question

We are given a set $T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K}$

Could you help me prove that if

$ \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in \mathbb{K}^{T^p}$

where $\rho: T^p \ni (x_1, ..., x_p) \rightarrow det [f_i(x_j)]_{i,j = 1, ... p} \in \mathbb{K}$

then $(\mathbb{K}^{T^p}, \phi)$ is the $p$-th exterior power of $\mathbb{K}^T$?

I know that $\phi$ is $p$-linear and anti-symmetric, because $\det$ is $p$-linear and anti-symmetric, but I have problems finding the unique linear map which makes the proper diagram commute.

Could you help me with that?

Thank you.