Multilinearity of the determinant

Leibniz formula. For an $n\times n$ matrix with columns $C_1,\ldots,C_n$, each entry $a_{ij}$ is the $i$-th entry of $C_j$. The Leibniz formula is
$$\det A = \sum_{\sigma\in S_n}\varepsilon(\sigma)\prod_{i=1}^n a_{\sigma(i),i}.$$

Linearity in column $j$. In this formula, column $j$ appears only through the factor $a_{\sigma(j),j}$ (the entry in row $\sigma(j)$ and column $j$). Since $a_{\sigma(j),j}=\lambda a'_{\sigma(j),j}+\mu a''_{\sigma(j),j}$, the product is linear in that entry, hence the entire sum is linear in column $j$.

Conclusion. The determinant is linear in each column separately (multilinear). Combined with the alternating property (zero when two columns are equal, sign change under transposition), this characterises the determinant.