In this paper,we present a construction of complex orthogonal design for space-time block codes in any number of antennas.Our construction achieves both the maximal rate and minimal delay.So far as we know,our construction is the first explicit-form construction,which has asymptotically optimal time complexity and space complexity.And new representation of complex orthogonal design might bring advantages in analyzing some properties.In addition,when the number of antennas n ≡ 1,2,3(mod 4),our construction satisfies transceiver signal linearization property,which allows for design of low complexity channel equalizers and interference suppressing filters.