That chain is sitting quite well on the front ring, almost perfectly, as far as can be discerned in the photo. In particularly, it doesn't appear to be elongated at all, which is usually the telltale sign of chain wear. When a chain is elongated, you can easily tell on the front ring, because it has so many teeth that the cumulative error of the elongated links causes it not to sit properly all the way around the ring.
I'm guessing you (or a previous owner) replaced the chain, possibly more than once, but didn't swap the rear cog at the same time.
That's a mistake.
Chains elongate as they wear, and at the same time, they wear the cog, and are worn by the cog. A chain which wore together with the cog might not skip, yet if you change to a new chain, that chain will skip on that old cog.
I've never been able to change a chain alone, without skipping cogs. (That is, the few times that I tried, before I stopped doing that).
Do check the tension! But note that chains don't have that much tension in derailleur systems (just that tension provided by the tension wheel) yet they don't skip, when in the proper condition. Basically there has to be ridiculously low tension. It's hard to tell from your picture, but it does look as if the chain is coming from the ring in a fairly straight line, no sag. Anyway, when you're pedaling hard, there is tension in the top section of the chain, between the ring teeth and the cog. Under this tension, they stay engaged even if the returning-to-ring section of the chain is sagging. If you don't believe me, think about this: when you remove a cassette from a hub on a derailleur system, you open a lock ring, for which you require a chain whip to hold the hub steady. That chain whip is just a piece of chain connected to a lever, loosely wrapped one of the cogs in the cassette: no tension in it at all, other than the applied tension. Nothing pulls on the unattached end of the chain to keep it wrapped around the cog! Yet, it grips tenaciously, enough to remove a stubborn lock ring. I've never had a chain whip skip, even when ridiculous force was needed (enough to break a spoke).
Get a new cog!