Antisymmetrized covector derivative is tensorial ($(dA{)}_{ab}={\partial }_a{A}_b-{\partial }_b{A}_a$)

For a smooth one-form $A$, the components ${\partial }_a{A}_b-{\partial }_b{A}_a$ transform as a covariant rank-two tensor. The non-tensorial second derivatives in the ordinary covector derivative under nonlinear coordinate changes are symmetric in $a,b$ and cancel. For a torsion-free connection the same components are ${\nabla }_a{A}_b-{\nabla }_b{A}_a$. This is the coordinate expression of $dA$, without choosing a metric.