The exterior algebra is a quotient of the Tensor algebra, with product the Wedge product

It can be defined as $\wedge^* V = T(V)/I(V)$

where $T(V)$ is the Tensor algebra of $V$ and $I(V)$ is the Ideal generated by elements of the form $v \otimes v$ where $v \in V$. So $I ( V )$ consists of all sums of multiples by $T ( V )$ on the left and right of these generators. We can also see $I(V)$ as the kernel of a certain projection map related to Permutations (see wave-notebook..)

Equivalently, we can define $\wedge^k V$ (called exterior power) to be the Vector subspace of $\otimes^k V$ on which the Symmetric group acts antisymmetrically (see wave-notebook..).

See here and wave-notebook.

If $v_1,...,v_n$ is a basis for $V$, then $\{v_{i_1} \wedge v_{i_2} \wedge ... \wedge v_{i_k} : 1 \leq i_1 < i_2 < ... i_k \leq n\}$ is a basis for $\wedge^k V$. Hence the dimension of $\wedge^k V$ is $\binom{n}{k}$. It is an associative supercommutative graded $\mathbb{R}$-algebra

– The wedge product, exterior power and exterior algebra can be defined for Vector bundles too. The elements of an exterior power of vector bundles are called Exterior forms