Am I right to think that gene is just set/group of alleles that is formed judging from the product of these alleles?

And allele being the section of DNA that codes for specific product.

For example, we have phenotypic product - hair color. There are alleles responsible for hair color. And therefore we group these alleles into set called “gene X” based on the premise of them being responsible for product “hair color”.

In other words reasoning goes as follows: alleles responsible for hair color are to be grouped in set called “gene X”

Correct me please, I think I am somewhere wrong in my reasoning.