  The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a
$\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called
Kauffman relations. In this article, for any closed oriented surface $\Sigma$
we provide an explicit spanning family for the skein modules $S(\Sigma\times
S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their
question about the dimension of $S(\Sigma\times S^1)$ being $2^{2g+1} + 2g -1$.
