Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A(\Omega)=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ is planar, $\mathbf A$ is parallel to its normal and has magnitude equal to the area of $\Omega$.
As it turns out, Stokes' theorem implies that $\mathbf A$ depends only on the boundary of $\Omega$; any other surface with the same boundary and orientation has the same directed area. So really, we are talking about directed areas of oriented closed curves in $\mathbb R^3$. Given an oriented closed curve $\Gamma$, we can construct any surface whose boundary is $\Gamma$ and compute its directed area. For convenience we may take the surface to be a cone with its vertex at the origin, and thereby obtain $$\mathbf A(\Gamma)=\frac12\oint_\Gamma\mathbf r\times\mathrm d\mathbf r.$$
It seems like this should be a useful quantity. It determines, for example, the flux enclosed by the curve due to any constant vector field $\mathbf v$ as $\mathbf v\cdot\mathbf A$. But I have never seen it discussed in any text. Does it have a standard name, and is it ever useful in practice?
It's called vector area; a vector whose components are given by signed area of the coordinate projections of the surface.
In addition to the reference given on Wikipedia, see Linear Algebra via Exterior Products by Sergei Winitzki (p.72-73). Also, there is a related question How can area be a vector? on Physics.SE.