In reading the ArXiv paper referenced in the post, the authors are using matrices with complex entries. (Near the beginning they talk about generators of SO(2), one of them being a square root of -I.) If one is looking for complex matrices in an orthogonal group SO(n), one can choose a scaled version of a complex Hadamard matrix of order n, where n is any positive integer, to get a matrix with entries having the same norm (absolute value), as well as having a row and a column having all entries equal to 1/(scale value, which may be n or square root of n).
(For the section being considered having display (62), it is unclear to me if the authors restrict themselves to matrices with real entries. For the purposes of the question, it seems to me that using matrices with complex entries is appropriate for carrying out their analysis, and that restricting the order to accommodate real Hadamard matrices is unnecessary.)
Gerhard "Complex Numbers Are Numbers Two" Paseman, 2020.02.03.