Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby $n \in \mathbb{C} \setminus \{ 0, 1, \ldots, 2 k - 2 \}$ so that $ \mathbb{C}A_k(n)$ is semisimple. How could a basis $M_{k} = M$ of $ \mathbb{C}A_k(n)$ be constructed so that $M$ is closed under the multiplicative operation on $\mathbb{C}A_k(n)$, in such a way so that $M$ is a monoid under this operation, and how could a product rule for elements in $M$ be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis $M$ of the desired form using Halverson and Ram's matrix unit construction for partition algebras, Benkart and Halverson's bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras.