Orthogonality of single-particle wavefunctions in Slater determinant

From the definition of Slater determinant it seems that the set of single-particle wavefunctions is chosen to be a orthonormal one. Is orthogonality required for the Slater determinant to describe an antisymmetric N-electron wavefunction?

As Lorents answer correctly points out, it is not necessary (to use orthonormal single-particle wavefunctions), but just much more convenient. In addition, it is no real restriction, since $$\left|\begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{matrix}\right|=\left|\begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1)+c_{12}\chi_1(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1)+c_{1N}\chi_1(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2)+c_{12}\chi_1(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2)+c_{1N}\chi_1(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N)+c_{12}\chi_1(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N)+c_{1N}\chi_1(\mathbf{x}_N) \end{matrix}\right|$$ and more generally for $$C=\begin{bmatrix} c_{11} & c_{12} & \cdots & c_{1N} \\ c_{21} & c_{22} & \cdots & c_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ c_{N1} & c_{N2} & \cdots & c_{NN} \end{bmatrix}$$ we have $$\left|\begin{bmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{bmatrix}C\right|=\left|\begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{matrix}\right|\operatorname{det}(C)$$ Hence we can orthogonalize the wavefunctions without changing the resulting many-particle state. (This also shows that the specific basis of the subspace is not important, as long as one takes care of the phase factor.)