Skylake wasn't a revolutionary improvement, it was evolutionary. Even a Haswell i3 still has a 50-60% single-threaded advantage. The i3 4160 came with a 3.6ghz stock clock, which would roughly equal a 5.8ghz FX CPU in single-threaded performance. You can get fairly close with a healthy overclock - a 4.8-5ghz FX-6300 would not be significantly slower - but to get there, you'll have to spend some money on a motherboard, power supply, and cooler.
Hyperthreading is worth an average of 30% more throughput, which puts a stock i3 4160 at roughly the multithreaded performance of a 3ghz FX-6300, taking module penalty into account. So, at stock, an FX-6300 has a multithreaded advantage, but a lot less than you'd think considering it has 6 cores.