The problem of maximizing secrecy rate of multiple-input multiple-output multiple-eavesdropper (MIMOME) channels with arbitrary numbers of antennas at each node is studied in this paper. First, the optimization problem corresponding to the secrecy capacity of the MIMOME channel is converted to an equivalent optimization based on Givens rotations and eigenvalue decomposition of the covariance matrix. In this new formulation, precoder is a rotation matrix which results in a positive semi-definite (PSD) covariance matrix by construction. This removes the PSD matrix constraint and makes the problem easier to tackle. Next, a Broyden-Fletcher-Goldfarb-Shanno (BFGS)-based algorithm is developed to find the rotation and power allocation parameters. Further, the generalized singular value decomposition (GSVD)-based precoding is used to initialize this algorithm. The proposed rotation-BFGS method provides an efficient approach to find a near-optimal transmit strategy for the MIMOME channel and outperforms various state-of-the-art analytical and numerical methods. In particular, the rotation BFGS precoding achieves higher secrecy rates than the celebrated GSVD precoding, with a reasonably higher computational complexity. Extensive numerical results elaborate on the effectiveness of the rotation-BFGS precoding. The new framework developed in this paper can be applied to a variety of similar problems in the context of multi-antenna channels with and without secrecy.