All nurses in Sweden are sjuksköterska, regardless of gender, and you cannot call them sjukskötare. In Finland however, all nurses are sjukskötare instead.
So in Standard Swedish, it is incorrect to call a male nurse sjukskötare.
This word is an exception. We generally use the -are forms for all genders, they are not considered masculine. But sjuksköterska is an old feminine form that has stuck.
Pretty bad in English too. The word "nurse" comes from the Latin nutrix, with a clearly feminine ending, meaning a wetnurse, i.e. a woman breastfeeding another’s child. As a verb, to "nurse" a child still means to breastfeed. It’s related to "nourish" and "nutrition". Its extension to women in other caring roles is recent. Its extension to men, even more recent.
Apparently "sköterska" also means "nurse." What is the distinction between it and "sjuksköterska?" Is a sjuksköterska a hospital/clinic nurse (i.e. a nurse for sick people, as in the name) whereas a sköterska is a general term for a caring professional (e.g. long term care nurses, homecare assistants, etc.)?
It is very similar to Russian words for the medical attendants. Санитарка ('a woman who takes care of the sick', 'aid-woman') usually refers to a nurse who works at a hospital or a clinic, and санитар ('a man who takes care of the sick', 'aid-man') refers to a male nurse who works at a mental health clinic or on an ambulance.
Moreover, we have three more words for 'nurse'. Медсестра (or медбрат for men) is rather a paramedic than just a nurse (but we usually translate it as 'nurse' - it is an old Soviet tradition :)), сиделка is very similar to санитарка, but it refers to one who works at the sick's home and for informal fees, and нянечка is very close to санитарка, but is more informal and tender. All that nouns (except for санитар and медбрат) are feminine.
At Swedish hospitals there are three main professions except doctors, sjuksköterska as above, and undersköterska, who also has an education for this, but a shorter one. And the third profession is vårdbiträde, who doesn't necessarily have any formal education. There are also more specific terms, like ambulanssjuksköterska who works in ambulances, operationsjuksköterska who works in surgery, etc. All those have special education for their fields.
sköterska and sjuksyster on the other hand are informal terms that don't have a very specific meaning and are not job titles.
As Zmrzlina said, skötare is short for mentalskötare, which is also a specific title. You can't change the gender in any of the job titles, the same is form is used for everyone.
Also, different terms are used in Finland.
Yes, the elderly might do that. But in any case it's relatively rare and shouldn't be used in talking to a nurse, since they will often prefer to be called their actual professional title.
According to a friend of mine who works as a nurse, they use syster or syrra (slang) internally among themselves, but that's another thing.