quantum groups - Basis for Universal Calculus

You can describe $Omega_A=ker(m:Aotimes Ato A)$ as the quotient of the free $A$-bimodule generated by symbols $d(a)$, one for each element $ain A$, by the sub-bimodule generated by the elements of the form $$d(ab)-d(a)\,b-a\,d(b), qquad a,bin A,$$ together with the elements of the form $$d(lambda 1), qquad lambdain k$$ with $k$ being the base field.
The elements ${a d(b):a,bin A}$, when seen in $Omega_A$, span $Omega_A$ over $K$ but are not linearly independent over $k$.

To extract from this a $k$-basis of $Omega_A$ you need to know more than a basis of $A$. For example, if you know a presentation of $A$ given by generators and relations, you can obtain a basis using essentially Groebner bases.