Actually, I'll jump in on this one. You totally can, as a core rule, without a Feat. It's called "attacking with an Improved Weapon". You're not proficient (without Tavern Brawler or some other such), and it does 1d4 damage (a one-handed improvised weapon - probably bludgeoning, but you might try an edge-on attack for slashing?). Also, it's an attack as part of an Attack Action, not a Bonus Action. [DM call on whether you can off-hand / TWF it.]
Shield Master essentially makes you trained in a variation of this, if you do it with along an attack as a combo. [Of course, then I also quickly house rule that the Shield Master feat makes you proficient in this shield bash option.]