In general, a volume element is a k-form on an K-dimensional manifold. a k-form w on $\mathbb{R}^{n}$ is defined as
$w(x) = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}}(x) \cdot\mathrm{d} x_{i_{1}}\wedge \mathrm{d} x_{i_{2}}\wedge...\wedge \mathrm{d} x_{i_{k}}$
For cylindrical coordinates, the line element is a 1-form in $\mathbb{R}^{3}$, and $\mathrm{ds}= \mathrm{dr}\hat{\mathrm{r}} + \mathrm{dz}\hat{\mathrm{z}}+\mathrm{r}\mathrm{d}\phi \hat{\phi}$. I can understand how this follows from the definition given above, since the line element is a 1-form. But I don't understand where the factor r in front of the $\mathrm{d}\phi \hat{\phi}$ comes from, using the definition of the differential form?