[Bearkiller72]

You can just write them down. Ya know with pencil and paper.

You might also do it, like our Sq.Ldr. does: check the eventlog.lst. You'll find it in your game installation folder.
Could be you need to activate the logging process in conf.ini:
[game]
Arcade=0
HighGore=1
mapPadX=0.65902776
mapPadY=0.4425926
viewSet=57
Intro=0
NoSubTitles=0
SubTitlesLines=3
NoChatter=0
NoHudLog=0
NoLensFlare=0
iconTypes=3
eventlog=eventlog.lst
eventlogkeep=1
3dgunners=1
HakenAllowed=1
ClearCache=1

You can open and edit eventlog.lst with any text editor. Listed in it you'll find the who, when and where you want to know about:
e.g.
[19.08.2010 20:41:56] Mission: net/dogfight/TEST/RAF_UK_BoB_1c1_test.mis is Playing

[20:41:56] Mission BEGIN

[20:44:54] Pilotname:HurricaneMkIIc(0) seat occupied by Pilotname at 200301.55 245497.66

[20:45:46] Pilotname:HurricaneMkIIc in flight at 200580.67 244546.56

[20:48:03] Bf-110C-4B shot down by Pilotname:HurricaneMkIIc at 196343.84 239373.03

[20:50:47] Pilotname:HurricaneMkIIc landed at 200328.11 246032.47
[20:50:53] Pilotname entered refly menu
[...]

That would be considered a "proven kill" by our Sq.Ldr., because the Pilot returned to base.