The determinant can be defined **axiomatically** as the unique function $\det: M_n(\mathbb{F}) \to \mathbb{F}$ satisfying:
1. **Multilinearity**: linear in each row (column)
2. **Alternating**: swapping two rows changes sign
3. **Normalization**: $\det(I) = 1$

These three properties uniquely characterize the determinant and can be used to derive all other properties.