In this letter we design a new family of space-time block codes (STBC) for multi-input multi-output (MIMO) systems. The complex orthogonal STBC achieves full diversity and full transmission rate with fast maximum-likelihood decoding when only two transmit antennas are employed. By combining the Alamouti STBC and the multidimensional signal constellation rotation based on the cyclotomic number field, we construct cyclotomic orthogonal space-time block codes (COSTBCs) which can achieve full diversity and full rate for multiple transmit antennas. Theoretical analysis and simulation results demonstrate excellent performance of the proposed codes, while the decoding complexity is further reduced.