To say "area is a pseudo-vector" you need to qualify a couple of things.
First that you're working in 3-space. (otherwise they are at best bi-vectors = rank two tensors)
Second that you're talking about oriented area elements (which are negative when you reverse orientation) typically defined by two vectors as the parallelogram they define.
The thing to remember is that "vectors" and "scalars" and such are so defined for arbitrary linear transformations on the space not just for rotations. A pseudo-scalar transforms like a volume element, being unchanged by rotations but scaling by the determinant of a more general transformation matrix. Since a vector defining a line element plus an area element together define a volume element (parallelepiped = oblique box) the area element must be a pseudo-vector (in 3 dim).
The vector associated with an area element is the normal vector of length numerically equal to the area. Then dotting a vector with this "pseudo-vector" gives the "pseudo-scalar" volume.
The types and dimensions follow the combinatorics of the the binomial coefficients. In 3 dim you have: 1,3,3,1 corresponding to scalar,vector, bivector=pseudo-vector, pseudo-scalar.
In four dimensions you would have: 1, 4, 6, 4, 1 and so:
scalar, vector, bivector(=pseudo-bivector), trivector=pseudo-vector, pseudo-scalar.
In five dimensions you would have 1,5, 10, 10, 5, 1 and so ...
