Let $Cl_n$ be the Clifford algebra (over reals) $$ Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle. $$ There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear maps $$ Cl_n \to \operatorname{Hom}_{K}(W,W) $$ e.g. for $K=\mathbb{R},\mathbb{C},\mathbb{H}.$ For instance, there is a unique irreducible $\mathbb{C}$-representation of $Cl_{n=2m}$ with $W=\mathbb{C}^{2^m}$ and two inequivalent $\mathbb{C}$-irreps of $Cl_{n=2m+1}$ with the same $W$.
The question is whether I could infer the classification of Clifford algebras themselves from the representations, e.g. by using the Artin-Wedderburn theorem. Is Artin-Wedderburn directly applicable, so that the following is true?
$$ \begin{align} Cl_{n=2m}\otimes_{\mathbb{R}}\mathbb{C} &\simeq \operatorname{Mat}_{2^m}(\mathbb{C}) , \\ Cl_{n=2m+1}\otimes_{\mathbb{R}}\mathbb{C} &\simeq \operatorname{Mat}_{2^m}(\mathbb{C}) \oplus \operatorname{Mat}_{2^m}(\mathbb{C}). \end{align} $$
If this is the case, one may expect that the full information of $Cl_n$ should come from its $\mathbb{R}$-irreps. Is it true?