You can describe $\Omega_A=\ker(m:A\otimes A\to A)$ as the quotient of the free $A$-bimodule generated by symbols $d(a)$, one for each element $a\in A$, by the sub-bimodule generated by the elements of the form $$d(ab)-d(a)\\,b-a\\,d(b), \qquad a,b\in A,$$ together with the elements of the form $$d(\lambda 1), \qquad \lambda\in k$$ with $k$ being the base field. The elements $\{d(a):a\in A\}$, when seen in $\Omega_A$, 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.

You can describe $\Omega_A=\ker(m:A\otimes A\to A)$ as the quotient of the free $A$-bimodule generated by symbols $d(a)$, one for each element $a\in A$, by the sub-bimodule generated by the elements of the form $$d(ab)-d(a)\\,b-a\\,d(b), \qquad a,b\in A,$$ together with the elements of the form $$d(\lambda 1), \qquad \lambda\in k$$ with $k$ being the base field. The elements $\{a d(b):a,b\in A\}$, 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.