Crossing over the fan is not the best, if you can gently bend it to persuade to go around the fan. But that is not critical.
You've raised an interesting point. Many current mobo's have two or three fan connectors, each identified a little differently. Almost all have at least a CPU Fan and a general case fan connector. Some also have one marked PSU Fan. It is important to hook up the right fan to the right connector, because most have three leads, and one of them is a signal back from the fan to the BIOS on the mobo. It monitors each fan speed separately AND uses different temperature sensors to control the fans' speeds. If you do what you started with - plug the CPU fan into the case fan connector - the BIOS will use the case temperature sensor line to control the the output to this connector, and it will actually be manipulating the CPU fan when it's trying to cool the general case. Worse, and this one's VITAL, it will have NO signal coming back from a fan to indicate the speed of the CPU fan - it was connected elsewhere! - and it will shut down your system quickly to prevent burning out your CPU because its fan is not running!
Make sure the right fan is plugged into the right mobo connector. And if you do NOT have a special lead coming out of the PSU to plug into the PSU fan connector, don't worry. Many PSU's do not. All that one does is allow the BIOS to monitor the PSU's fan speed, but it does not try to control it or go into a panic if it is not working. It assumes the PSU is taking care of its own temperature control.