On the one hand what you're suggesting is rather easy to do. Take a complex manifold $X$ of complex dimension $n$ (so real dimension $2n$). The set of "things that look like $f dz_1 \wedge \cdots \wedge dz_n$ locally" is a very important object, called the canonical line bundle of $X$ and denoted $K_X$. As Didier noted, the things in $K_X$ ("the sections of $K_X$") are differential $n$-forms and can be integrated over submanifolds of $X$ of real dimension $n$.
Unfortunately this is not a super interesting thing to do if you're a complex geometer, because if $n$ is even and you can find a complex submanifold of (real) dimension $n$ in $X$, the integral of anything in $K_X$ over it will be zero. The reason is essentially one of linear algebra, but vaguely speaking to get a nonzero integral over a complex manifold you need the same number of $dz$ and $d\bar z$ in your form, and we only have $dz$'s here.
"But wait," you say, "I integrate holomorphic forms like $f dz$ in the plane over paths all the time and I love it."
That is true, but turns out to be a rather specific feature of complex dimension $1$. There a form like $f dz$ can be integrated over a real path, and closed such paths are the boundaries of open complex submanifolds, so Stokes' theorem applies. By Cauchy's theorem, the real money in integrating holomorphic forms is really in integrating meromorphic forms, that is forms with poles. In higher dimensions the poles of a meromorphic form are no longer isolated and can't be contained inside the boundary of a real hypersurface (and even less inside a lower-dimensional real manifold).
What we can do in higher dimensions is to take two holomorphic $n$-forms $\alpha$ and $\beta$ and try to integrate $\alpha \wedge \overline{\beta}$ over $X$, as that will be a $2n$-form on a (real) $2n$-dimensional manifold. This leads to things like Bergman kernels and Weil-Peterson metrics, which are both active areas of research. There is also some kind of theory of residues in higher dimensions, but it is less developed and important than in complex dimension one.
Best Answer
Terminology aside, the idea is that we can define a contravariant functor $$\Omega^*:(\text{Smooth Manifolds})\to (\text{Chain Complexes of Vector Spaces})$$ by assigning to a manifold $M$ its de Rham Complex $$ \Omega^0(M)\xrightarrow{d}\Omega^1(M)\xrightarrow{d}\Omega^2(M)\xrightarrow{d}\cdots\xrightarrow{d} \Omega^n(M). $$ You can find the definition of a functor between categories formally defined all over (like on wikipedia) but I'll sketch the idea here. Being a (contravariant) functor means that $\Omega^*$ assigns to each manifold a chain complex like above, and to each smooth map of manifolds $f:M\to N$ an associated morphism of chain complexes $$\Omega^*(f):\Omega^*(N)\to \Omega^*(M).$$ The contravariance indicates the above flipping of directionality of the maps, i.e. $\Omega^*(f)$ goes from $\Omega^*(N)$ to $\Omega^*(M)$ rather than the other way around. More has to be true: $\Omega^*$ should respect compositions in that $\Omega^*(f\circ g)=\Omega^*(g)\circ \Omega^*(f)$ and have $\Omega^*(\Bbb{1}_M)=\Bbb{1}_{\Omega^*(M)}$. Anyway, so much for the formalities. The picture here is quite simple: we have specified $\Omega^*(M)$ explicitly already, and associated to $f:M\to N$ we define $\Omega^*(f)$ to be the morphism of chain complexes $f^*$ as $\require{AMScd}$ \begin{CD} \Omega^0(N) @>d>> \Omega^1(N)@>d>>\cdots@>d>>\Omega^n(N)\\ @V f^* V V @V f^*V V @VV V @V f^* VV\\ \Omega^0(M) @>>d> \Omega^1(M)@>>d>\cdots@>>d>\Omega^n(M). \end{CD} Note that the diagram commutes by $f^*\circ d=d\circ f^*$. You can check the functoriality properties listed above.