Let $V$ be a complex vector space with a nondegenerate quadratic form $\langle -, -\rangle$. Let $\mathrm{Cl}(V)$ be the Clifford algebra: the quotient of the tensor algebra $\mathrm{T}(V)$ by the relations $v \otimes v = \langle v, v \rangle 1$.
If $V$ is even dimensional, say $\dim(V) = 2k$, I have read that $\mathrm{Cl}(V)$ has a unique (up to isomorphism) irreducible representation $S$ of rank $2^k$, and an algebra isomorphism $\mathrm{Cl}(V) \cong \mathrm{End}_\mathbb{C}(S)$, basically a matrix algebra. If $V$ is odd-dimensional, say of dimension $2k+1$, instead $\mathrm{Cl}(V) \cong \mathrm{End}_{\mathbb{C}}(S) \oplus \mathrm{End}_{\mathbb{C}}(S)$, so $S$ arises in two different non-isomorphic ways.
Is there a coordinate-free definition of the $2^k$-dimensional irreducible representation $S$ of $\mathrm{CL}(V)$? Something expressed directly in terms of tensor operations and the quadratic form?
Each source I have looked at (including previous math.SE questions) either omits/glosses over the construction of this representation, or else uses a choice of basis and explicit coordinates thereafter (making it hard for me to tell what parts are natural, if any). That said, I wouldn't be surprised if I'm just overlooking something – if so, a link would be much appreciated.