The short answer is unfortunately no.
Solar systems automatically shut down when they detect grid power to be down. This is mandated by utility companies for safety reasons.
Even if this wasn't the case, solar production increases with direct sunlight and can go up and down quite rapidly depending on the weather and other factors. This would be unideal as it could cause appliance faults and you would only have power when your solar system was producing sufficient energy.
Only with a special type of Hybrid system will you be able to use energy produced from your solar system during a power outage.
A Hybrid system combines solar power from a PV system with another power generating energy source. In this case the other energy source is a battery and specifically an inverter with emergency power supply (EPS) capabilities. This is basically the ability for the battery to supply a specified set of EPS circuits in the event of a grid failure by isolating them from the mains grid.
These specified circuits are decided on by the home owner as the essential, low power appliances. These may include your fridge, modem, medical equipment, lights etc. The installer would advise you against putting more loads on those EPS circuits as most battery inverters will have an upper limit to the power they can supply as well as a capacity limit. Even with only the essential loads on your EPS circuits you will quickly drain a fully charged battery.
For example a standard hybrid may have max 5kW AC power output connected to a 6kWh battery which when fully charged will only last about an hour if you pull a constant 5kW load with the inverter.
These systems can be expensive. It is important to weigh up the cost vs the benefits.