The dual space is a very different animal in the general world of $R$-modules.

Review point: for an $R$-module $M$, the dual space $R*$ is the space of linear mappings from $M$ into $R$.

For finite-dimensional vector spaces, the dual space $V*$ is isomorphic to $V$. Once we choose a basis, we can immediately go back-and-forth, in a one-to-one manner, between vectors $v$ and their duals.