On smooth Riemannian manifolds, you can always find a smooth partition of unity: $$\{\varphi_\alpha\}_{\alpha\in\mathfrak{A}},\qquad \sum_{\alpha\in\mathfrak{A}}\varphi_\alpha=1$$ and only finitely many $\varphi_\alpha$ are non-zero in any region. If you calculate a local $n$-form from orthonormal coordinates: $$dx^1\land dx^2\land\cdots\land dx^n$$ You can use this to define a volume form $d$vol over your Riemannian manifold.
How is this done for analytic or Hermitian manifolds? Since, in this case, you cannot have a non-zero function with compact support, so a partition of unity is impossible. I'm sure there are results on this but I am struggling to find any myself. Any help/links would be awesome. Thanks!