A.~Einstein considered a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric and $F\ne0$ is skew-symmetric, and a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. M. Prvanovi\'c (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the so-called $f^2$-torsion condition.We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds $(M,f,g)$, satisfying the $f^2$-torsion condition, where $F(X,Y)=g(X,fY)$,give explicit formulas for the torsion in terms of $\nabla^g F$, $dF$ and a new (1,1)-tensor $\widetilde Q:=-f^2-{\rm Id}$, and show that in the almost Hermitian case, our main result reduces to the coordinate-free form of Prvanovi\'c's solution.Finally, we describe special Einstein connections, i.e., the difference tensor has the property~$K_XY=-K_YX$, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.