The Wikipedia development describes a process starting from an n-dimensional vector space V over a field K and a quadratic form Q for V.
The result is a new vector space C of 2^{n} dimensions.
There are within C two subspaces, one naturally isomorphic to V and another naturally isomorphic to K.
Much of Clifford algebra literature identifies both V and K with their natural counterparts in C.
This risks notation confusion: is αu the vector space multiplication of a scalar by a vector, or is it the new algebraic multiplication between two elements of C?
Another confusion is whether αβ is a field multiply, a scalar by vector multiply, or the multiply defined in the new algebra.
In short is αβ a construct in K, V or C?

You might have guessed that by the axioms the three meanings are all the same. I have never seen notation introduced in which you could prove that these syntactically ambiguous expression are not semantically ambiguous. It is reminiscent of the reals being honorary members of the complex. It is why these constructs are called hypercomplex numbers.

There are several developments on the web and the Wikipedia is the most abstract, requiring more preliminaries, but is also coordinate free. I will follow the more pedestrian and more common development here.

Choose a basis γ_{i} for V, 0≤i<n.
**New Axioms**: (1) if i≠j then γ_{i}γ_{j} = −γ_{j}γ_{i},

(2) γ_{i}γ_{i} = −1.

The axioms from associative algebras now define the product of any two vectors when we know the products of any two base vectors.
The γs are base vectors for V, not C.
What are base vectors for C?
We cannot prove from the axioms that γ_{1}γ_{2} ≠ γ_{3} but we seek the largest algebra generated by K and so we assume that the products of base elements of K are distinct except as required by the axioms.
Any product of base elements can be rewritten in increasing order in the subscripts.
That will cause some sign reversals.
There can be no repeated factors in this expression since γ_{i}γ_{i} = −1.
In short for each of the 2^{n} subsets of the set of n base vectors of V, there is exactly one base vector for C, those base elements from base vectors of V multiplied together.
This is a complete base set for C.
It also proves that C has 2^{n} dimensions.

Another was to define C_{n}, the Clifford algebra over V_{n} is to define C_{0} as the reals and to define C_{n} as an algebra of pairs: <u, v> where u, v ∊ C_{n−1}.
The rest of this is developed here.