I am perfectly happy with the skill system in Lamentations of the Flame Princess but I cannot help myself when I think alternative rules. I don't necessarily use those myself but I like to create. Here's an alternative skill system for LotFP what is similar to newer Dungeons & Dragons editions.
Difficulty Class (DC) is a target number minimum needed for the skill test for a success. DC is determined by the Referee according to the situation. DCs are:
10 - Routine
15 - Challenging
20 - Tough
25 - Hard
30+ - Extreme
Roll d20 and add skill score (1 to 6) and ability modifier. If the task is something character class is supposed to be good at add also ½ of the level (round down).
Formula: d20 + skill + ability modifier (+ ½ character's class)
If the result is higher than the DC the task is a success.
Natural 20 is always a successful result except in Extreme difficulty class
Natural 1 is always a botch. Character doesn't only fail but something unexpected and negative happens.