Photonic crystals exhibiting a photonic band gap in both TE and TM polarizations are particularly interesting for a better control of light confinement. The simultaneous achievement of large band gaps in both polarizations requires to reduce the symmetry properties of the photonic crystal lattice. In this letter, we propose two different designs of two-dimensional photonic crystals patterned in high refractive index thin silicon slabs. These slabs are known to limit the opening of photonic band gaps for both polarizations. The proposed designs exhibit large complete photonic band gaps: the first photonic crystal structure is based on the honey-comb lattice with two different hole radii and the second structure is based on a "tri-ellipse" pattern in a triangular lattice. Photonic band gap calculations show that these structures offer large complete photonic band gaps deltaomega/omega larger than 10% between first and second photonic bands. This figure of merit is obtained with single-mode slab waveguides and is not restricted to modes below light cone.