Grappling and shoving are the only RAW ways to "attack with" a skill
The Grappling rules state:
When you want to grab a creature or wrestle with it, you can
use the Attack action to make a special melee attack, a grapple. If
you're able to make multiple attacks with the Attack action, this
attack replaces one of them.
The target of your grapple must be no more than one size larger than
you and must be within your reach. Using at least one free hand, you
try to seize the target by making a grapple check instead of an attack
roll: a Strength (Athletics) check contested by the target's Strength
(Athletics) or Dexterity (Acrobatics) check (the target chooses the
ability to use). You succeed automatically if the target is
incapacitated. If you succeed, you subject the target to the grappled
condition. The condition specifies the things that end it, and you can
release the target whenever you like (no action required).
Escaping a Grapple. A grappled creature can use its action to
escape. To do so, it must succeed on a Strength (Athletics) or
Dexterity (Acrobatics) check contested by your Strength (Athletics)
check.
Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the
creature is two or more sizes smaller than you.
The rules on shoving a creature immediately follow the grappling rules:
Using the Attack action, you can make a special melee attack to shove
a creature, either to knock it prone or push it away from you. If
you're able to make multiple attacks with the Attack action, this
attack replaces one of them.
The target must be no more than one size larger than you and must be
within your reach. Instead of making an attack roll, you make a
Strength (Athletics) check contested by the target's Strength
(Athletics) or Dexterity (Acrobatics) check (the target chooses the
ability to use). You succeed automatically if the target is
incapacitated. If you succeed, you either knock the target prone or
push it 5 feet away from you.
Using grappling and shoving, you can make use of your Athletics skill in combat to prevent an enemy from moving and/or knock it prone to grant advantage to attacks from within 5 feet. However, there are no other ways, by RAW, to "attack with" a skill (whether Athletics or any other skill); all other attacks involve an attack roll instead of an ability check.