Can we introduce independent coordinates on a sphere such that any great circle could be represented as a linear equation (like line on the plane)? If yes, what is a generalization for higher dimensions? Thank you in advance.

$\begingroup$ When you say "multidimensional" are you then talking about great hypercircles (or whatever they would be called)? The intersection of a hyperplane with the sphere. $\endgroup$ – Gerald Edgar Sep 15 '17 at 11:58
This can be done locally by projection from the center of the sphere. Great circles go to lines, so if you pull back the cartesian coordinates from the plane to a hemisphere, every great halfcircle is described by a linear equation.
You cannot extend this beyond the hemisphere, because two great circles meet in two points, but a system of linear equations cannot have two solutions.