A simple test was proposed for testing independence of high-dimensional random normal vectors. The method relies on two steps: first, projecting the primary high-dimensional data onto a low-dimensional subspace several times using random projection matrices; second, constructing the test statistic by use of the classical statistics obtained over projected low-dimensional datasets. Simulations were conducted to compare the performance of the proposed test with the state-of-the-art existing ones, in terms of the sizes and powers of the tests. Results from our Monte Carlo study reveal that the suggested test performs well both in terms of the size and power. In the end, the proposed methodology was illustrated on two gene data sets, the Colon and Leukemia cancer datasets.