Whether or not you are charged for an individual bag going over the 20kg depends on the airline.
It seems to me that if the airline says "2 bags of 20kg" (rather than "up to 2 bags, max 40kg") that means that each bag cannot go over 20kg (without extra charge at least).
You should be able to find the answer in the luggage info, or terms & conditions, in the air company's website. Or at least, if you've enough time, by emailing their customer service.
My guess is that you could easily be charged for bringing only 1 bag, but when it is over 20kg.
You know that air companies usually charge by the kilogram, and this can be expensive, maybe 20 dollars a kilo, for long distance, maybe more 50 dollars or even 70 dollars or more.
My advice is that it is better to buy a small, cheap bag, for less than 20 dollars, which you may be able to use again, as a gym bag or overnight bag, and transfer some things into it in the airport. That way, you won't be paying for nothing.
Though, you should also make sure in the first place of the baggage terms - looking through the website, or contacting the company.
I remember, though, I've had great problems with this in the past.
Once I was flying with Lufthansa (Europe) from Munich to Sofia, Bulgaria. My luggage was too heavy, but it was actually impossible to work out this from the way the luggage terms were described in the booking info. When I booked, and packed, I assumed it meant one thing.
I flew from somewhere else to Munich and stayed a few days (with another airline). So, when it came time to go to the Munich flight, I went to the airport the day before because of weather and illness, my flight was changed. When I was at the airport I suddenly realised I may have interpreted the baggage info wrongly, and that it meant the opposite of what I thought, and that I may have to pay about $100 or more for extra luggage kilos.
So, I took my printout of the booking terms, and brought it to a female member of staff at the luggage desk. I said what I thought it meant originally, and she read it over and over again, and she agreed with me. Therefore, I was satisfied that I'd got it right first time, and I wouldn't have to pay. I returned the next day for my changed flight. The man at check-in told me that, because of the weather many flights were missed and the queues were so long, and he didn't have time to charge me, but normally my baggage with the ticket I had would be over $100 dollars extra to pay at check-in. He didn't charge me anything though, because of all the queues and confusion.
So, both I and the woman staff the day before HAD gotten it wrong first time. And she worked for Lufthansa.
On returning, Lufthansa had booked my return flight with their associate airline, Swiss Air, who's extra luggage terms were different to Lufthansa, and extra luggage prices MUCH more expensive.
I went through the terms on the Swiss Air website, and at least with Swiss there was no doubt I would have to pay, and quite a lot. But I never thought to check when booking a return flight with Lufthansa directly on their website, that the baggage terms would be different when returning.
It turned out Swiss's extra luggage costs were about $250 extra for the full flight back to the UK. Damn, damn, damn.
It can be very unclear - so always be prepared to do anything you can to pay less. Don't go for buying extra luggage on check-in because it can be so expensive, and you might not realise until there.
The best advice is to contact the airline with what you think you'll bring by email. TELL THEM IN THE EMAIL that you can't work out how the baggage terms applies in your situation. (Otherwise they'll just refer you to the terms, and they still may easily refer you back their terms again, even if you say you can't work out how they apply to you. I've had that happen to me before).
Then take a PRINTOUT of the airline's response, so you can bring it with you and reasonably expect to rely on it.
