Question: It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do better by defining a tensor as a multilinear map from $ V\times...\times V\times V^\ast\times ...\times V^\ast\to \mathbb{F} $, where $V$ is a vector space over $\mathbb{F}$ (often taken to be a tangent space). The transformation law then follows.

My current understanding of spinors feels like the first, dissatisfying definition: they're just "things that transform like spinors" - that is, they're elements of a vector space which transform according to a projective representation of $SO(n)$ which is genuinely multi-valued (i.e. it's not just a true representation of $SO(n)$). We could call this the "spinor transformation law". Note that this is something we've put in "by hand": the way a spinor transforms is not a property of some underlying object, but is built into our definition.

My question is: can we define spinors without reference to the way they transform, just as we did for tensors? Is there some object "underlying" the definition of spinors in terms of transformations, just as tensors are "really" multilinear maps?

\begin{align}
\text{Tensor Transformation Law}&\to \text{Tensors as multilinear maps}\\
\text{Spinor Transformation Law}&\to \text{??? }
\end{align}

Answer: 
The proper analogous formalization of spinors is not to view them as some sort of different functions from tensors on the same underlying vector space $V$, but instead to expand our idea of the underlying geometry: Where tensors are multilinear functions on vector spaces, tensors with "spinor" and "vector" parts are multilinear functions on [super vector spaces](https://en.wikipedia.org/wiki/Super_vector_space) $V = V\_0\oplus V\_1$ where the odd part $V\_1$ is a spinorial representation of $\mathrm{Spin}(V\_0)$. (nlab calls these spaces [super-Minkowski spacetimes](https://ncatlab.org/nlab/show/super+Minkowski+spacetime)).

Via the dual representation, the linear functions on $V\_1$ inherit a representation of the spin group. The (multi)linear functions also inherit the super-grading (a linear function that is zero on the odd part is even, and a linear function that is zero on the even part is odd), and purely even such functions are just ordinary tensors, and purely odd functiona are pure spinors.

Note that we still put in the spin representation $V\_1$ by hand - the choice is not determined by the base space $V\_0$. This is, in some way, not surprising - a notion of "spin" and spinor is genuinely *more* than just having a vector space: All (pseudo-Riemannian) manifolds (modeled on the vector spaces $\mathbb{R}^n$) have a notion of tensors built on tensor products of the (co)tangent spaces, but [not all manifolds have spinors](https://en.wikipedia.org/wiki/Spin_structure#Obstruction), i.e. the possibility to associate consistently a spinorial representation to every point of the manifold. For simple vector spaces the choice of a notion of spin is not obstructed, but it is still a choice.

That the supergeometric approach is nevertheless the "correct" (or at least a useful) one is seen when we turn to field theory, where one must represent fermionic/spinorial degrees of freedom by anti-commuting variables, and the $\mathbb{Z}/2$-grading of the underlying vector space then allows us to do this simply by declaring that the odd components anti-commute.
Yes. Spinors are elements of representation spaces of objects known as a [Clifford algebras](https://en.wikipedia.org/wiki/Clifford_algebra).

A Clifford algebra is basically a vector space turned into an algebra via the product rule

$$
v\cdot w=2g(v,w)\Bbb{1}
$$

where $g$ is some metric on the vector space itself. The most famous Clifford algebra is the Dirac algebra, i.e. the algebra of the Dirac matrices (for which the vector space is $\Bbb{R}^{4}$ and the metric is the Minkowski metric). If instead you use $\Bbb{R}^{3}$ as the base vector space, with the Euclidean metric, you obtain the Pauli algebra.

Once you have a Clifford algebra you can look for its representations (or "modules", as they are known in the literature). The elements of these representations are the [spinors](https://en.wikipedia.org/wiki/Clifford_algebra#Spinors). The spinors corresponding to $\Bbb{R}^{4}$ with the Minkowski metric are the Dirac spinors, whereas those corresponding to $\Bbb{R}^{3}$ with the Euclidean metrics are the spinors of $SO(3)$/$SU(2)$.