The fact that your ideals have two generators defines a surjection $R^{\oplus 2}\to I$. Now if you can compute $r$ generators for the kernel of that map (aka “syzygies” between your generators), this gives you an exact sequence $R^{\oplus r} \to R^{\oplus 2} \to I \to 0$ presenting $I$, which tensored by $I$ gives $I^{\oplus r} \to I^{\oplus 2} \to I\otimes_R I \to 0$ which then presents $I\otimes_R I$ (and hence $\bigwedge^2 I$ if you want) as a quotient of the free module $R^{\oplus 4}$ by an explicitly generated submodule. But I presently can't remember how we do that “if” part.
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