The axioms of matroids are (Wikipedia):
A finite matroid $M$ is a pair $(E,\mathcal{I})$, where $E$ is a finite set (called the ground set) and $\mathcal{I}$ is a family of subsets of $E$ (called the independent sets) with the following properties:
- $\mathcal{I}\neq\emptyset$
- $A'\subseteq A\in\mathcal{I}\Rightarrow A'\in\mathcal{I}$
- $A,B\in\mathcal{I}\wedge |A|>|B|\Rightarrow \exists x\in A:B\cup\{x\}\in\mathcal{I}\wedge |B\cup\{x\}|>|B|$
In linear algebra any vector $v\neq 0$ is independent, but an element of $E$ doesn't have to occur in any set in $\mathcal{I}$ according to the axioms.
It seems to me that generator sets for finitely generated modules over some ring would be a more general representation of the matroids? Where torsion elements corresponds to $v$ above.