Is there a way to find n mutually perpendicular vectors in a n-dimensional space, where coefficient of each direction is either 1 or -1?
For example, when n = 2, a possible solution could be:
A = (1, -1), B = (1, 1)
Similarly, when n = 4, a possible solution could be: A = (-1, 1, 1, 1), B = (1, -1, 1, 1), C = (1, 1, -1, 1), D = (1, 1, 1, -1)
Is there some way to calculate a solution for higher values of n?
The $n$ vectors you are looking for form what is called a Hadamard matrix. See that article for various ways to construct Hadamard matrices and a discussion of what is known about the existence of Hadamard matrices.