My cologne is causing mild redness and stinging on my neck and inner elbow. How can I manage this?
I think you are suffering from irritation rather than an allergy, given that the redness is mild, appears only where you apply it and doesn’t spread. An allergy is more serious and you would need to stop using your cologne.
Presuming that this is an irritation, then it depends on how much you like your cologne and how irritating is the irritant! If you badly want to continue to wear it, you might ask the manufacturer if it comes in a different formulation, perhaps in an oil base or as a balm, as these could be gentler.
The two areas you mention are where the skin tends to be thinner, so irritation is more likely. You could try applying a light oil or moisturiser before spraying the cologne as this would potentially offer your skin a layer of protection and actually may make the scent last for longer.
Cologne is usually formulated at a concentration for unshaved skin, whilst formulations for application to shaved skin must contain a significantly lower concentration of essential oils and aroma chemicals. I don’t expect you shave inside your elbows(!) but you may shave you neck and this would likely make skin much more sensitive.
Check the ingredients list, any citrus oils are quite likely culprits, as are many aroma chemicals, but alcohol can do it too. If the skin reaction worsens, you should probably stop using it and do keep an eye on it; if it looks like contact dermatitis you will probably need to avoid the irritant for good. But if the stinging is worth the smell, then maybe spray on your hair (the skin on your scalp is thicker, and hair will make skin contact minimal). For similar reasons, you could spray it on your forearms, not inside the elbow and the back of your hands will be less sensitive than your neck.
I hope it improves and those around you appreciate your suffering for their olfactory pleasure!