An orthonormal square Zernike basis set is generated from circular Zernike polynomial apodized square mask by use of the linearly independent set Gram-Schmidt orthogonalization technique. Based on the concepts of inner product, Euclidean space and norm in the linear algebra, a standard Zernike polynomial set is made orthogonal and a new orthonormal basis of polynomials named Z-square polynomial is established. Wavefront data in square aperture can be fitted with our new orthonormal set. It can not only fit the wavefront data with Z-square basis set itself, but also can be linearly composed of standard Zernike basis set by linear reverse transform and endows the decomposed wavefront modes with a correspondent aberration meaning. The experimental results show that the Z-square polynomial set can fit the wavefront aberration data in lens design efficiently and can also fit the practical wavefront phase data of Hartmann-Shack wavefront sensor testing, it provides a method of wavefront data analysis.