Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that right? 



Questions on calculating volume using n-1 forms
The question is a bit vague so I will try to make the best of it. Suppose that $(M, g)$ is a connected, noncompact oriented, $n$-dimensional Riemann manifold and $dV\_g\in\Omega^n(M)$ is the associated volume form. Suppose that $D\subset M$ is an open, precompact subset of $M$ with smooth boundary $\partial D$. Since $M$ is noncompact we have $H^n(M,\mathbb{R})=0$ and we deduce that there exists an $(n-1)$ form $\eta$ such that $d\eta=dV\_g$. Stokes' theorem then implies that


$$ \mathrm{vol}(D, g)=\int\_D dV\_g = \int\_{\partial D} \eta. $$


For example, if $(M,g)$ is the Euclidean space $\mathbb{R}^n$, then we can take


$$\eta=\frac{1}{n}\sum\_{k=1}^n (-1)^{k-1} x^k dx^1\wedge \cdots \wedge\widehat{dx^k}\wedge \cdots dx^n. $$


If $M$ is compact, and $x\in M\setminus \bar{D}$, then $M\setminus x$ is noncompact and arguing as above we can find $\eta\_x\in \Omega^{n-1}(M\setminus x)$ such that $dV\_g|\_{M\setminus x}=d\eta\_x$ and 


$$ \mathrm{vol}(D, g)= \int\_{\partial D} \eta\_x. $$


