Testing for normality has always been a center of practical and theoretical interest in statistical research. In this paper a test statistic for bivariate normality is proposed. The underlying idea is to investigate all the possible linear combinations that reduce to the standard normal distribution under the null hypothesis and compare the order statistics of them with the theoretical normal quantiles. The suggested statistic is invariant with respect to nonsingular matrix multiplication and vector addition. We show that the limit distribution of an approximation to the suggested statistic is represented as the supremum over an index set of the integral of a suitable Gaussian Process. We also simulate the null distribution of the statistic and give some critical values of the distribution and power results.