Let $V$ be an infinite-dimensional vector space over a field $K$. Then it is known that $\dim V < \dim V^*$. More precisely, by a result attributed to Kaplansky and Erdos, we have $\dim V^* = |K|^{\dim V}$.

I have not seen an actual construction of a basis of $V^*$. My question is: given a basis $B$ of $V$, is there an explicit description of a basis of $V^*$ in terms of $B$? Can you do this at least in the case where $\dim V$ is countable?