That's not the definition of small.
The big issue is airplane. You have to take out the video card. Outside of extremely expensive boards with metal reinforcements, that flimsy plastic PCI-E port is going to snap. If you're using a non stock cooler, it can't be big / heavy otherwise that may cause issues with the motherboard.
The other issue on plane, is static. Airplanes are dry, you can easily build up a static discharge on accident on the plane, I've done it many many times.
You really want to go with an mitx board, case (they usually support two slot) and an SFX PSU. Even then it's not safe for your equipment.
You're looking at something like this with the suggested. Going by dimension you can fit it underneath the seat infront, however do not put it in the overhead.
PCPartPicker part list /
Price breakdown by merchant
Motherboard: Asus P8H61-I R2.0 Mini ITX LGA1155 Motherboard ($73.99 @ SuperBiiz)
Case: Silverstone RVZ01B Mini ITX Desktop Case ($78.99 @ SuperBiiz)
Power Supply: Corsair SF 450W 80+ Gold Certified Fully-Modular SFX Power Supply ($89.99 @ Amazon)
Total: $242.97
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-03-31 17:15 EDT-0400
If I may ask, why exactly are you going through this route as opposed to... anything else quite honestly, like a laptop.