This article tries to employ the notions of {\it periodicity}, {\it center of an algebra}, and {\it tensor product} to generate an isomorphic expression for the Clifford algebra $\scr{C}l_{p,q}$ of the real vector space $\Bbb{R}^{p, q}$ with a nondegenerate quadratic form of signature $(p, q)$, where $p$ and $q$ are non-negative integers. I feel that this work contributes to the knowledge in the discipline because of its attempt to create a relationship between the three notions and to obtain the matrix associated to the Clifford algebra $\scr{C}l_{p,q}$ which is described in terms of the quaternions.