Summary: In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian \[ \begin{cases} (-\Delta)^su(x)=u^{m_1}(x)v^{n_1}(x), \quad & u(x)\geq 0,\, x\in\mathbb{R}_+^n,\\ (-\Delta)^sv(x)=u^{m_2}(x)v^{n_2}(x), & v(x)\geq 0,\,x\in\mathbb{R}_+^n,\\ u(x^{\prime},-x_n)=-u(x^{\prime},x_n), & x=(x^{\prime},x_n)\in\mathbb{R}^n,\\ v(x^{\prime},-x_n)=-v(x^{\prime},x_n), & x=(x^{\prime},x_n)\in\mathbb{R}^n, \end{cases} \] where \(0 < s < 1\), \(m_i, n_i > 0_{(i=1,2)}\), \(n > 2s\), \(\mathbb{R}_+^n =\{(x^{\prime},x_n)|x_n>0\} \). We first show that the solutions only depend on \(x_n\) variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to \(x_n\) variable (for the critical and subcritical cases \(m_i+n_i\leq \frac{n+2s}{n-2s}\,_{(i=1,2)}\) in the \(\mathcal{L}_{2s}\) space). Furthermore, when \(m_1=n_2=p\), \(n_1=m_2=q \), in the cases \(p+q+2s\geq 1 \), we obtain a Liouville theorem for the cases \(p+q\leq \frac{n+2s}{n-2s}\) in the \(\mathcal{L}_{2s}\) space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain \(\Omega \). Using the anti-symmetric property of the solutions, one can extend the space from \(\mathcal{L}_{2s}\) to \(\mathcal{L}_{2s+1} \), we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions.