The vector potential generated by an infinitesimal current element is proportional to $\mathrm d\vec l/r$, where $\mathrm d\vec l$ is the infinitesimal displacement along the wire and $r$ is the distance to the point at which the vector potential is evaluated. Note that you got this the wrong way around in the question; you described $r$ and not $\mathrm d\vec l$ as a vector.
The magnetic field is the curl of the vector potential. Since only $r$ and not $\mathrm d\vec l$ depends on the point at which the vector potential is evaluated, the magnetic field generated by the infinitesimal current element is proportional to
$\vec\nabla\times\frac{\mathrm d\vec l}{r}=\left(\vec\nabla\frac{1}{r}\right)\times\mathrm d\vec l=\frac{\vec r}{r^3}\times\mathrm d\vec l\;.$
If you want to calculate the magnetic field at the centre of a circular current loop, the vector product $\vec r\times\mathrm d\vec l$ is the same everywhere and points out of the plane of the loop, since $\vec r$ and $\mathrm d\vec l$ are always at right angles to each other, so the magnetic field points out of the plane of the loop and its magnitude is proportional to $\frac R{R^3}(2\pi R)=\frac{2\pi}R$.
If you want to read up on this in more detail, this set of notes might be helpful.