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Denote by \({\mathcal{C}\ell_{p,q}}\) the Clifford algebra on the real vector space \({\mathbb{R}^{p,q}}\). This paper gives a unified tensor product expression of \({\mathcal{C}\ell_{p,q}}\) by using the center of \({\mathcal{C}\ell_{p,q}}\). The main result states that for nonnegative integers p, q, \({\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}\) where \({p + q \equiv \varepsilon}\) mod 2, \({\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}\) mod 8 and \({\delta = \lfloor i / 4 \rfloor}\).