Which computer operating system are you using-- Windows, Mac, or iOS?
Anvil Studio is a free DAW for Windows-- that is, the basic program (which includes the functionality you need) is free, but there are also a number of "premium" features which must be purchased as "accessories" if you want to use them. Anvil Studio doesn't filter out SysEx messages, so it would be a good choice as a first DAW.
GarageBand is a free DAW for Mac and iOS-- again, the basic program is free, but certain things must be purchased if you want to use them. You can record MIDI input, but I'm not certain whether you can save ("export") a project to a MIDI file, and I don't know whether GarageBand filters out SysEx messages.
Aria Maestosa is a free DAW/sequencer for Mac and Windows (and Linux/Unix-- but since Yamaha doesn't make a USB-MIDI driver for Linux/Unix, I won't cover it). I've only just downloaded and installed it, so I'm not familiar with it yet.
Cockos REAPER isn't free, but the trial version is free, it isn't crippled, and it continues to work after the trial period ends. It's available for both Mac and Windows. I'm not very familiar with it yet.
PreSonus Studio One has a free version that's available for Mac and Windows.
Zynewave Podium is another DAW that has a free version, which is available for Windows.
There are many other DAWs out there, and the "best" ones tend to be rather expensive, although there are also some great low-priced DAWs. For instance, I'm a big fan of
Acoustica Mixcraft, which is a low-priced DAW for Windows-- although Mixcraft does filter out SysEx messages, which is unfortunate. Anyway, most of the commercial DAWs let you download free trials, which can help you decide which one you want to buy.