Specifically, a k-blade is a k-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade k. In detail:[1] A vector subspace of finite dimension k may be represented by the k-blade formed as a wedge product of all the elements of a basis for that subspace.
When the space is endowed with a volume form (an alternating k-multilinear scalar-valued function), such a k-blade may be normalized to take unit value, making the correspondence unique up to a sign.
In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space that is distinct from regular scalars.
In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars.
Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.