Here’s a thing. Beyond the Wall follows the old-school logic regarding any kind of skill check, which is to try to roll under one of the six attributes with a d20. This is a fine and elegant way of handling the skill system.
However… it’s a bit counter-intuitive because the other two times you roll a d20 require a roll high. Even the BtW core book acknowledges this.
So, here are two alternative ways to make skill checks. Both use d6 rather than d20 to make rolls. Doing so separates the rolls vs. threats (saves and to-hit) from the more freeform skill checks.
The first one is borrowed from Greg Saunder’s Summerland. The player takes an number of dice and tries to roll under the appropriate stat. The GM sets the difficulty of the task by the number of dice rolled. 2 dice is going to be a fairly easy task (average of 7 should fit under most stats), 3 dice will be medium to challenging and 4 dice will be quite hard.
Skills (and help) function as per the rulebook, giving a bonus to the stat under test.
The WaRP Method
WaRP uses 2 dice for an untrained character, 3 dice for skilled and 4 dice for exceptional rolls. Taking that example add 2d6 to the stat for an unskilled roll; for a skill, add 3d6 (or 4d6 if the character has doubled up on the skill).
For a simple skill test, the target is 15; increase the difficulty by increments of 5 (or 3 for more granularity) for something harder.
WaRP also handles advantages and disadvantages with a bonus or penalty die; an extra die is rolled and the worst number (bonus) or best number (penalty) is dropped. The bonus die might work for giving help and any time the GM wants to shade the roll in the player’s favour.