Awk - output the second line of a number of .dat files to one file

Remove while loop and make use of shell brace expansion and also FNR, a built-in awk variable:

awk 'FNR==2{print $0 > "output.dat"}' file{1..80}.dat

What about ... head -n 2 input.dat | tail -n 1 | awk ...

sed would be enough:

sed -sn 2p file{1..80}.dat > output.dat

-s option is needed to print 2nd line from each file, otherwise only 2nd line of first file will be printed.