Note that this is not a great solution, an Uno is not fit for the job. Take a look at the Arduino Leonardo, for example, it supports MIDI over USB natively (no special firmware required) and has a hardware UART you can use for hardware MIDI out.
If you cannot find a Leonardo, there are other boards with the same ATMEGA32U4 processor. For example, the Arduino Micro, the SparkFun Pro Micro, and the Adafruit Itsy Bitsy 32u4.
Thanks a lot guys, I got myself an Arduino Pro Micro with the AtMega32u4, it was really easy to make it a MIDI controller using MIDIUSB Library, now the only thing I need is to send MIDI OUT thru the Tx pin at the same time.
Any idea how to do that ? I have connected a 5 pin with 220ohm resistor, etc. I need some help with the code.