The Kauffman bracket skein module S(M) of a 3-manifold M is a Q(A)-vector space spanned by links in M modulo the so-called Kauffman relations. For any closed oriented surface Sigma we provide an explicit spanning family for the skein modules S(Sigma x S-1). Combined with earlier work of Gilmer and Masbaum (Proc. Amer. Math. Soc. 147 (2019) 4091-4106), we answer their question about the dimension of S(Sigma x S-1) being 2(2g+1) + 2g-1.