Create different subsets of variables in a matrix with no repetition

Question

I have a matrix of about 20 different variables. My scope is to create different combinations of these variables (no repetition, each of min 3 variables) and store them each in a new data frame. I tried with combn and expand.grid but it takes as inputs all the values connected to each variable; while I want it to take only the "name" of the variable and not its values. Thanks in advance