Seeking its basis elements under definition for free R modules

Question

I'm given with notion of free $R$ module that its definition is:
An abelian group $ <F,+> $ is free module on $ A \subset F $ if for all non zero $ x\in F $ $ x $ is uniquely expressed in terms of finite $R$-combinatons generated by $A$.

But we can define free $R$ module in correspondence with arbitrary set $A $ s.t. $Free_R(A) = \{ \sum_{i\leq n}{(r_i,a_i)}|n \ finite \}$ and by not allowing any extra relationship between its elements except it being abelian it is evident that it satisfies definition of free R module as above.

But I can't see existence of $A$ in $Free(A)$ of fairly expressible form, as any attempts like $(1_R,a_i)$ brings in unfair assumption that refuses pure uniquessness structure in $Free(A)$, what is going on?