A Giambelli formula for classical $G\mathrm{/}P$ spaces

Let $G$ be a classical complex Lie group, $P$ any parabolic subgroup of $G$, and $G\mathrm{/}P$ the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in the cohomology ring of $G\mathrm{/}P$ as a polynomial in certain special Schubert class generators. Our formula extends to one that applies to the torus-equivariant cohomology ring of $G\mathrm{/}P$ and to the setting of symplectic and orthogonal degeneracy loci.