As @mhoran_psprep described, employers will withhold 6.2% of wages until the annual wage limit is reached, then additional wages will not have Social Security Tax withheld. As far as I know, the tax regulations don't allow them to withhold it by "averaging" it out over the year as you have described.
But, even if they could, it's not likely that they would, because there is no guarantee that you will continue at the same wage for each and every month.
For example, using your figure of $20,000.00 per month...
Suppose that after 5 months, you left the company for another job, retired, or took an unpaid leave for the remainder of the year. Again using your figures, they would have withheld $612.25 * 5 = $3,061.25. But for those 5 months, you were paid $20,000.00 * 5 = $100,000.00. So, for your $100,000 in wages, they should have withheld $6,200.00 but they only withheld $3,061.25, leaving a shortage of $3,138.75.
After that (still using the $20,000.000 monthly wages figure), for every additional month that you work for that company (beyond the 5 in this example), the shortage would be reduced, but even at 10 months (with 2 months off), the shortage would still be $1224.50. Unless they could recover this shortage from you, the company would be liable for paying it (plus perhaps some interest and penalty).
The only way the monthly withholding would work out to $612.25 * 12, is if your monthly wages were (legitimately) $9,875.00, and then in December, you received an additional bonus of $121,500.00 (assuming there was no fraud involved).