<<xAct`xCoba`

DefManifold[M4,4,IndexRange[a,f]]

DefMetric[-1,metricg[-a,-b],Cd,{",","∂"},PrintAs->"g"]

DefTensor[v[a],M4]

Cd[a]@v[-b]

IndexCoefficient[Cd[a]@v[-b],Cd[-c]@v[d]]

IndexCoefficient[Cd[-b]@v[a],Cd[-c]@v[d]]

IndexCoefficient[(Cd[a]@v[-b]-Cd[-b]@v[a]),Cd[-c]@v[d]]

(Cd[a]@v[-b]-Cd[-b]@v[a])/2

IndexCoefficient[(Cd[a]@v[-b]-Cd[-b]@v[a])/2,Cd[-c]@v[d]]

IndexCoefficient[Expand[(Cd[a]@v[-b]-Cd[-b]@v[a])/2],Cd[-c]@v[d]]//Simplification