A 44-year-old man has been arrested on suspicion of murder following the attack in front of shoppers on Thursday afternoon.
Customers and colleagues tried to help by bravely pinning down the attacker before police arrived.
Police have not named the woman but said her family have been informed.
Skipton mayor Andy Rankine said:
"Skipton is a small but close-knit community where most people know each other. We are in shock about the sad events of yesterday where a store worker was murdered whilst going about her duties. The whole town is in shock and grieving over the loss."
People left messages on a community Facebook page, saying the married victim was a mother, and that "she was such a happy, friendly person", while another described her as a "bubbly, lovely girl".
One wrote: "Horrendous, so tragic for her family and for those who witnessed this barbaric act."
Another customer said she had seen the Aldi employee earlier that day wearing a Christmas jumper on her shift.
One shopper who was in the store at the time of the attack said she had "never been so scared in my life".
The witness, who asked to remain anonymous, heard "loads of screams" then saw the woman lying on the floor and the man pinned down as "everyone screamed and ran up and down".
She said: "I just saw the aftermath; I was so scared I ran off.
"All the staff were racing about, not knowing what to do."
Paramedics were also called to the store in Keighley Road but the woman died despite their efforts to save her.
North Yorkshire Police said the incident was not terror-related or believed to be a hate crime.
A police spokesman said:
"We are not in a position to identify the victim at this stage, and we urge people to respect her family's privacy and to avoid speculation and rumour on social media."
Witnesses and anyone with information are asked to call 101.
People affected by the traumatic scene can seek support from the force's Major Incident Response Team by calling 07974 745194.