Does there exist a matrix $\mathbf{A}$ that takes any vector $\mathbf{v}\in \mathbb{R}^n$ into the circulant matrix $\mathbf{C}_{\mathbf v} = \mathbf{A}\mathbf{v} \in \mathbb{R}^{n\times n}$ constructed from $\mathbf{v}$?
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The map is $$ L(a_1\,\ldots,a_n) = \sum_{ i=1}^n a_i L\mathbb{e}_i, $$ with $ L \mathbb{e}_i \in \mathbb{R}^{n\times n} $ given by $$ (L\mathbb{e}_i)\mathbb{e}_q\cdot \mathbb{e}_p = \begin{cases} 1 &\text{ if } q=p \text{ mod } n,\\ 0 &\text{otherwise} \end{cases}. $$ To represent it into a matrix, you need to decide how you order the $n^2$ basis vectors of $\mathbb{R}^{n\times n}$. But if you just want to explicitly write the linear dependence on the $a_i$, representing $L$ as above is possibly more practical.
